Spanning treeFrom Wikipedia, the free encyclopedia

Contents

1 Agreement forest 11.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Agreement forests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2.1 Rooted case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 Acyclic agreement forests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Optimization problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Arborescence (graph theory) 32.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3 Arboricity 53.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.2 Arboricity as a measure of density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.3 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.4 Related concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4 Aronszajn tree 84.1 Existence of -Aronszajn trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.2 Special Aronszajn trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.3 Construction of a special Aronszajn tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

5 Bethe lattice 105.1 Relation to Cayley graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.2 Lattices in Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

i

ii CONTENTS

6 Block graph 126.1 Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126.2 Related graph classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136.3 Block graphs of undirected graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

7 Borvkas algorithm 157.1 Pseudocode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157.2 Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167.4 Other algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

8 Branch-decomposition 188.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188.2 Relation to treewidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198.3 Carving width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198.4 Algorithms and complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198.5 Generalization to matroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208.6 Forbidden minors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

9 Bridge Protocol Data Unit 239.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

10 Cantor tree 2410.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

11 Capacitated minimum spanning tree 2511.1 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

11.1.1 Esau-Williams heuristic[2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2511.1.2 Sharmas heuristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

11.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2511.3 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2611.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

12 Caterpillar tree 2712.1 Equivalent characterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2712.2 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2812.3 Enumeration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2812.4 Computational complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2812.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2812.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

CONTENTS iii

12.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

13 Cayleys formula 3013.1 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3013.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3013.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

14 Centered tree 3214.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3214.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3214.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

15 Christodes algorithm 3415.1 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3415.2 Approximation ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3415.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3515.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

16 Circuit rank 3616.1 Related concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3616.2 Almost trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3616.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3716.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

17 Degree-constrained spanning tree 3817.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3817.2 NP-completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3817.3 Degree-constrained minimum spanning tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3817.4 Approximation Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3817.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

18 Distributed minimum spanning tree 4018.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4118.2 MST in message-passing model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4118.3 GHS algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

18.3.1 Preconditions[1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4118.3.2 Properties of MST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4218.3.3 Description of the algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4218.3.4 Progress property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

18.4 Approximation algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4418.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

19 Euclidean minimum spanning tree 4519.1 Lower bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

iv CONTENTS

19.2 Algorithms for computing EMSTs in two dimensions . . . . . . . . . . . . . . . . . . . . . . . . . 4619.3 Higher dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4619.4 Subtree of Delaunay triangulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4719.5 Expected size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4819.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4819.7 Planar realization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4819.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4819.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

20 Expected linear time MST algorithm 5020.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

20.1.1 Borvka Step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5020.1.2 F-heavy and F-light edges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

20.2 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5120.3 Correctness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5120.4 Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

20.4.1 Random Sampling Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5120.4.2 Expected Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5220.4.3 Worst case analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

20.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5320.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

21 Game tree 5421.1 Solving game trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

21.1.1 Deterministic Algorithm Version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5521.1.2 Randomized Algorithms Version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

21.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5621.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5621.4 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

22 Graphic matroid 5722.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5722.2 The lattice of ats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5722.3 Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5822.4 Matroid connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5822.5 Minors and duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5822.6 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5922.7 Related classes of matroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5922.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

23 HalpernLuchli theorem 6123.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

CONTENTS v

24 Heavy path decomposition 6324.1 Decomposition into paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6324.2 The path tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6324.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6324.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

25 Honest leftmost branch 6525.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6525.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

26 JechKunen tree 6626.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

27 K-ary tree 6727.1 Types of k-ary trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6727.2 Properties of k-ary trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6727.3 Methods for storing k-ary trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

27.3.1 Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6727.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6827.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6827.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

28 k-minimum spanning tree 6928.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7128.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

29 K-tree 7229.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

30 Kinetic minimum spanning tree 7430.1 General case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7430.2 H-minor-free graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7430.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7430.4 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

31 Kirchhos theorem 7531.1 An example using the matrix-tree theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7531.2 Proof outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7531.3 Particular cases and generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

31.3.1 Cayleys formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7731.3.2 Kirchhos theorem for multigraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7731.3.3 Explicit enumeration of spanning trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7731.3.4 Matroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

31.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

vi CONTENTS

31.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7831.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

32 Kruskals algorithm 7932.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8032.2 Pseudocode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8032.3 Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8032.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8132.5 Proof of correctness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

32.5.1 Spanning tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8132.5.2 Minimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

32.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8132.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

33 Kruskals tree theorem 8333.1 Friedmans nite form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8333.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8433.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8433.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

34 Kurepa tree 8534.1 Specializing a Kurepa tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8534.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8534.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

35 Laver tree 8735.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8735.2 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8735.3 Amoeba forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8735.4 Cohen forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8735.5 Grigorie forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8835.6 Hechler forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8835.7 JockuschSoare forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8835.8 Iterated forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8835.9 Laver forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8835.10Levy collapsing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8835.11Magidor forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8935.12Mathias forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8935.13Namba forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8935.14Prikry forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8935.15Product forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9035.16Radin forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9035.17Random forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

CONTENTS vii

35.18Sacks forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9035.19Shooting a fast club . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9035.20Shooting a club with countable conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9035.21Shooting a club with nite conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9135.22Silver forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9135.23References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9135.24External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

36 Level ancestor problem 9236.1 Naive methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9236.2 Jump pointer algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9236.3 Ladder algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

36.3.1 Stage 1: long-path decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9336.3.2 Stage 2: extending the long paths into ladders . . . . . . . . . . . . . . . . . . . . . . . . 9436.3.3 Stage 3: combining the two approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

36.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9436.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

37 Lowest common ancestor 9537.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9537.2 Extension to directed acyclic graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9637.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9637.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9637.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9637.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

38 Millikens tree theorem 10038.1 Strong embedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10038.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

39 Minimum bottleneck spanning tree 10139.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

39.1.1 Undirected graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10139.1.2 Directed graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

39.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10139.3 Camerinis algorithm for undirected graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

39.3.1 Pseudocode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10239.3.2 Running time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10239.3.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

39.4 MBSA algorithms for directed graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10239.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

40 Minimum degree spanning tree 104

viii CONTENTS

41 Minimum spanning tree 10541.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

41.1.1 Possible multiplicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10641.1.2 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10641.1.3 Minimum-cost subgraph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10641.1.4 Cycle property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10641.1.5 Cut property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10741.1.6 Minimum-cost edge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10741.1.7 Contraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

41.2 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10741.2.1 Classic algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10741.2.2 Faster algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10741.2.3 Linear-time algorithms in special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . 10841.2.4 Decision trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10841.2.5 Optimal algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10941.2.6 Parallel and distributed algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

41.3 MST on complete graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10941.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11041.5 Related problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11041.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11141.7 Additional reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11341.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

42 Minimum spanning tree-based segmentation 11642.1 Motivation for graph-based methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11642.2 From images to graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11642.3 Minimum spanning tree segmentation algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . 11642.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11742.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

43 Path graph 11843.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11843.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11843.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

44 Polytree 11944.1 Related structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12044.2 Enumeration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12044.3 Sumners conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12044.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12044.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12044.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

CONTENTS ix

44.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

45 Prims algorithm 12245.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12245.2 Time complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12345.3 Proof of correctness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12345.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12445.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12445.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

46 Prfer sequence 12846.1 Algorithm to convert a tree into a Prfer sequence . . . . . . . . . . . . . . . . . . . . . . . . . . 128

46.1.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12846.2 Algorithm to convert a Prfer sequence into a tree . . . . . . . . . . . . . . . . . . . . . . . . . . 12846.3 Cayleys formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12946.4 Other applications[3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13046.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13046.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

47 Random minimal spanning tree 13147.1 First model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

48 Random tree 13248.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

49 Rectilinear minimum spanning tree 13349.1 Properties and algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

49.1.1 Planar case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13449.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

49.2.1 Electronic design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13449.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13449.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

50 Rectilinear Steiner tree 13550.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13550.2 Computational complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

50.2.1 Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13550.2.2 Approximations and heuristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

50.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

51 Recursive tree 13851.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13851.2 Bijections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13851.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

x CONTENTS

51.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

52 Reverse-delete algorithm 14052.1 Pseudocode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14052.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14052.3 Running time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14052.4 Proof of correctness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

52.4.1 Spanning tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14152.4.2 Minimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

52.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14252.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

53 Serres property FA 14353.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14353.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

54 Shortest total path length spanning tree 14554.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

55 Shortest-path tree 14655.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14655.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

56 Spanning tree 14756.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14856.2 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

56.2.1 Fundamental cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14856.2.2 Fundamental cutsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14856.2.3 Spanning forests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

56.3 Counting spanning trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14956.3.1 In specic graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14956.3.2 In arbitrary graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14956.3.3 Deletion-contraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14956.3.4 Tutte polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

56.4 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15156.4.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15156.4.2 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15156.4.3 Randomization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15156.4.4 Enumeration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

56.5 In innite graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15256.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

57 Spanning Tree Protocol 15457.1 Protocol operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

CONTENTS xi

57.1.1 Data rate and STP path cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15857.1.2 Bridge Protocol Data Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15857.1.3 Bridge Protocol Data Unit elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

57.2 Evolutions and extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16157.2.1 Rapid Spanning Tree Protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16157.2.2 Per-VLAN Spanning Tree and Per-VLAN Spanning Tree Plus . . . . . . . . . . . . . . . 16357.2.3 Rapid Per-VLAN Spanning Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16357.2.4 VLAN Spanning Tree Protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16357.2.5 Multiple Spanning Tree Protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16357.2.6 Shortest path bridging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

57.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16457.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16557.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

58 Star (graph theory) 16758.1 Relation to other graph families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16758.2 Other applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16858.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

59 Steiner tree problem 16959.1 Euclidean Steiner tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17059.2 Rectilinear Steiner tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17159.3 Generalization of minimum Steiner tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17159.4 Approximating the Steiner Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17259.5 Steiner ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17259.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17259.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17259.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17359.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

60 Stemmatics 17460.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17460.2 Basic notions and objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17460.3 Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17660.4 Eclecticism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

60.4.1 External evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17860.4.2 Internal evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17860.4.3 Canons of textual criticism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17860.4.4 Limitations of eclecticism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

60.5 Stemmatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17960.5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17960.5.2 Limitations and criticism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

xii CONTENTS

60.6 Copy-text editing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18160.6.1 McKerrows concept of copy-text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18260.6.2 W. W. Gregs rationale of copy-text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18260.6.3 GregBowersTanselle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

60.7 Cladistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18660.8 Application of textual criticism to religious documents . . . . . . . . . . . . . . . . . . . . . . . . 186

60.8.1 Qur'an . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18760.8.2 Book of Mormon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18860.8.3 Hebrew Bible . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18860.8.4 New Testament . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19060.8.5 Talmud . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

60.9 Classical texts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19360.10Legal protection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19360.11Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19360.12See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

60.12.1 Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19460.12.2 Critical editions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19560.12.3 Lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

60.13Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19660.14References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20060.15Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20160.16External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

60.16.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20260.16.2 Bible . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

61 Strahler number 20361.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20361.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

61.2.1 River networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20461.2.2 Other hierarchical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20461.2.3 Register allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

61.3 Related parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20561.3.1 Bifurcation ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20561.3.2 Pathwidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

61.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20661.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20661.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

62 Suslin tree 20862.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20862.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

CONTENTS xiii

63 Tree (descriptive set theory) 20963.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

63.1.1 Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20963.1.2 Branches and bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20963.1.3 Terminal nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

63.2 Relation to other types of trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20963.3 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21063.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21063.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

64 Tree (graph theory) 21164.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

64.1.1 Plane tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21264.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21264.3 Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21264.4 Enumeration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

64.4.1 Labeled trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21364.4.2 Unlabeled trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

64.5 Types of trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21464.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21464.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21464.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21564.9 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

65 Tree (set theory) 21665.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21665.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21665.3 Tree (automata theory) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

65.3.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21765.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21765.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21865.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

66 Tree decomposition 21966.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21966.2 Treewidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22066.3 Dynamic programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22166.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22166.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22166.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

67 Tree spanner 22367.1 Known Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

xiv CONTENTS

67.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

68 Trmaux tree 22468.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22468.2 In nite graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22568.3 In innite graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22568.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

69 Uniform spanning tree 22769.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22769.2 The uniform spanning tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22769.3 The Laplacian random walk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22869.4 Grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

69.4.1 High dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22969.4.2 Two dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22969.4.3 Three dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

69.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23069.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

70 Uniform tree 23170.1 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

71 Unrooted binary tree 23271.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23271.2 Related structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

71.2.1 Rooted binary trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23371.2.2 Hierarchical clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23371.2.3 Evolutionary trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23371.2.4 Branch-decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

71.3 Enumeration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23471.4 Alternative names . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23471.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23471.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

72 Variation (game tree) 23672.1 Principal variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23672.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23772.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

73 Virtual Link Trunking 23873.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23873.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

74 WedderburnEtherington number 240

CONTENTS xv

74.1 Combinatorial interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24074.2 Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24174.3 Growth rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24174.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24174.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24274.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24274.7 Additional reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

75 Wiener connector 24475.1 Problem denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24475.2 Relationship to Steiner tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24475.3 Computational complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

75.3.1 Hardness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24575.3.2 Exact algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24575.3.3 Approximation algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

75.4 Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24575.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24575.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24675.7 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 247

75.7.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24775.7.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25275.7.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

Chapter 1

Agreement forest

In the mathematical eld of graph theory, an agreement forest for two given (leaf-labeled, irreductible) trees is any(leaf-labeled, irreductible) forest which can, informally speaking, be obtained from both trees by removing a commonnumber of edges.Agreement forests rst arose when studying combinatorial problems related to computational phylogenetics, in par-ticular tree rearrangements.[1]

1.1 PreliminariesRecall that a tree (or a forest) is irreductible when it lacks any internal node of degree 2. In the case of a rooted tree(or a rooted forest), the root(s) are of course allowed to have degree 2, since they are not internal nodes. Any tree (orforest) can be made irreductible by applying a sequence of edge contractions.An irreductible (rooted or unrooted) tree T whose leaves are bijectively labeled by elements of a set X is called a(rooted or unrooted) X-tree. Such a X-tree usually model a phylogenetic tree, where the elements of X (the taxonset) could represent species, individual organisms, DNA sequences, or other biological objects.Two X-trees T1 and T2 are said to be isomorphic when there exists a graph isomorphism between them whichpreserves the leaf labels. In the case of rooted X-trees, the isomorphism must also preserves the root.Given a X-tree T and a taxon subset Y X, the minimal subtree of T that connects all leaves in Y is denoted byT(Y). When T is rooted, then T(Y) is also rooted, with its root being the node closest to the original root of T. ThisT(Y) subtree needs not be a Y-tree, because it might not be irreductible. We therefore further dene the restrictedsubtree T|Y, which is obtained from T(Y) by suppressing all internal nodes of degree 2, yielding a proper Y-tree.

1.2 Agreement forestsAn agreement forest for two unrooted X-trees T1 and T2 is a partition {X1, X2, ..., X} of the taxon set X satisfyingthe following conditions:

1. T1|X and T2|X are isomorphic for every i {1,2,...,k} and2. the subtrees in { T1(X) : i {1,2,...,k} } and { T2(X) : i {1,2,...,k} } are vertex-disjoint subtrees of T1 and

T2, respectively.

The set partition {X1, X2, ..., X} is identied with the forest of restricted subtrees F = {T|X1, T|X2, ..., T|X},with either T=T1 or T=T2 (the choice of it begin irrelevant because of condition 1). Therefore an agreement forestcan either be seen as a partition of the taxon set X or as a forest (in the classical graph-theoretic sense) of restrictedsubtrees.The size of an agreement forest is simply its number of components. Intuitively, an agreement forest of size k fortwo phylogenetic trees is a forest which can be obtained from both trees by removing (k-1) edges in each tree andsubsequently suppressing internal nodes of degree 2.

1

2 CHAPTER 1. AGREEMENT FOREST

1.2.1 Rooted case

1.2.2 Acyclic agreement forestsA ranement on the above denition can bemade, resulting in the concept of acyclic agreement forest. An agreementforest F for two X-trees T1 and T2 is said to be acyclic if each of its tree components can be numbered in such a waythat if the root of one component X F is an ancestor of the root of another component X F in either T1 or T2,then the number assigned to X is lower than the number assigned to X .Another characterization of acyclicity in agreement forest is to consider the directed graph GF that has vertex set Fand a directed edge (X, X) if and only if i j and at least one of the two following conditions hold:

1. the root of T1(X) is an ancestor of the root of T1(X) in T12. the root of T2(X) is an ancestor of the root of T2(X) in T2

The directed graph GF is called the inheritance graph associated with the agreement forest F, and we call F acyclicif GF has no directed cycle.

1.3 Optimization problemsA (rooted, unrooted, acyclic) agreement forest F for T1 and T2 is said to be maximum if it contains the smallestpossible number of elements (i.e. it has the smallest size). In this context, it is the agreement between the two treeswhich is maximized: it explains why computing a maximum agreement forest actually means minimizing its numberof components. This leads to two dierent (but related) optimization problems. In both cases, we choose to minimize|F| 1 rather than |F|, because the former corresponds to the number of cuts to be done in each tree in order to obtainF.

maximal maximum unrooted MAF corresponds to TBR rooted MAF corresponds to rSPR acyclic MAF corresponds to HYB

AFs can be dened on non-binary trees AFs can be dened on more than two trees acyclic agreement forests have a role to play in the computation of HYB on 3 or more trees, but the relationshipis much weaker than in the case of 2 trees

Complexity FPT algorithms Approximation algorithms Exponential time algorithms

1.4 Notes[1] Jotun Hein; Tao Jiang; Lusheng Wang; Kaizhong Zhang (1996). On the complexity of comparing evolutionary trees.

Discrete Applied Mathematics 71 (1-3): 153169. doi:10.1016/S0166-218X(96)00062-5.

Chapter 2

Arborescence (graph theory)

In graph theory, an arborescence is a directed graph in which, for a vertex u called the root and any other vertexv, there is exactly one directed path from u to v. An arborescence is thus the directed-graph form of a rooted tree,understood here as an undirected graph.[1][2]

Equivalently, an arborescence is a directed, rooted tree in which all edges point away from the root; a number of otherequivalent characterization exist.[3][4] Every arborescence is a directed acyclic graph (DAG), but not every DAG isan arborescence.An arborescence can equivalently be dened as a rooted digraph in which the path from the root to any other vertexis unique.[1]

The term arborescence comes from French.[5] Some authors object to it on grounds that it is cumbersome to spell.[6]There is a large number of synonyms for arborescence in graph theory, including directed rooted tree[2][6] out-arborescence,[7] out-tree,[8] and even branching being used to denote the same concept.[8] Rooted tree itself hasbeen dened by some authors as a directed graph.[9][10][11]

Furthermore, some authors dene arborescence as to be a spanning directed tree of a given digraph.[11][12] The samecan be said about some its synonyms, especially branching.[12] Other authors use branching to denote a forest ofarborescences, with the latter notion dened in broader sense given at beginning of this article,[13][14] but a variationwith both notions of the spanning avor is also encountered.[11]

Its also possible to dene a useful notion by reversing all the arcs the arborescence, i.e. making them all point to theroot rather than away from it. Such digraphs are also designated by a variety of terms such as in-tree[15] or anti-arborescence[16] etc. W. T. Tutte distinguishes between the two cases by using the phrases arborescence divergingfrom [some root] and arborescence converging to [some root].[17]

The number of rooted trees (or arborescences) with n nodes is given by the sequence:

0, 1, 1, 2, 4, 9, 20, 48, 115, 286, 719, 1842, 4766, 12486, ... (sequence A000081 in OEIS).

2.1 See also Edmonds algorithm

Multitree

2.2 References[1] Gordon, Gary (1989). A greedoid polynomial which distinguishes rooted arborescences. Proceedings of the American

Mathematical Society 107 (2): 287. doi:10.1090/S0002-9939-1989-0967486-0.

[2] Stanley Gill Williamson (1985). Combinatorics for Computer Science. Courier Dover Publications. p. 288. ISBN 978-0-486-42076-9.

3

4 CHAPTER 2. ARBORESCENCE (GRAPH THEORY)

[3] Jean-Claude Fournier (2013). Graphs Theory and Applications: With Exercises and Problems. John Wiley & Sons. pp.9495. ISBN 978-1-84821-070-7.

[4] Jean Gallier (2011). Discrete Mathematics. Springer Science & Business Media. pp. 193194. ISBN 978-1-4419-8046-5.

[5] Per Hage and Frank Harary (1996). Island Networks: Communication, Kinship, and Classication Structures in Oceania.Cambridge University Press. p. 43. ISBN 978-0-521-55232-5.

[6] Mehran Mesbahi; Magnus Egerstedt (2010). Graph Theoretic Methods in Multiagent Networks. Princeton University Press.p. 38. ISBN 1-4008-3535-6.

[7] Ding-Zhu Du; Ker-I Ko; Xiaodong Hu (2011). Design and Analysis of Approximation Algorithms. Springer Science &Business Media. p. 108. ISBN 978-1-4614-1701-9.

[8] Jonathan L. Gross; Jay Yellen; Ping Zhang (2013). Handbook of Graph Theory, Second Edition. CRC Press. p. 116.ISBN 978-1-4398-8018-0.

[9] David Makinson (2012). Sets, Logic and Maths for Computing. Springer Science & Business Media. pp. 167168. ISBN978-1-4471-2499-3.

[10] Kenneth Rosen (2011). Discrete Mathematics and Its Applications, 7th edition. McGraw-Hill Science. p. 747. ISBN978-0-07-338309-5.

[11] Alexander Schrijver (2003). Combinatorial Optimization: Polyhedra and Eciency. Springer. p. 34. ISBN 3-540-44389-4.

[12] Jrgen Bang-Jensen; Gregory Z. Gutin (2008). Digraphs: Theory, Algorithms and Applications. Springer. p. 339. ISBN978-1-84800-998-1.

[13] James Evans (1992). Optimization Algorithms for Networks and Graphs, Second Edition. CRC Press. p. 59. ISBN 978-0-8247-8602-1.

[14] Bernhard Korte; Jens Vygen (2012). Combinatorial Optimization: Theory and Algorithms (5th ed.). Springer Science &Business Media. p. 18. ISBN 978-3-642-24488-9.

[15] Kurt Mehlhorn; Peter Sanders (2008). Algorithms and Data Structures: The Basic Toolbox. Springer Science & BusinessMedia. p. 52. ISBN 978-3-540-77978-0.

[16] Bernhard Korte; Jens Vygen (2012). Combinatorial Optimization: Theory and Algorithms (5th ed.). Springer Science &Business Media. p. 28. ISBN 978-3-642-24488-9.

[17] Tutte, W.T. (2001), Graph Theory, Cambridge University Press, pp. 126127, ISBN 978-0-521-79489-3

2.3 External links Weisstein, Eric W., Rooted Tree, MathWorld.

Chapter 3

Arboricity

The arboricity of an undirected graph is the minimum number of forests into which its edges can be partitioned.Equivalently it is the minimum number of spanning forests needed to cover all the edges of the graph.

3.1 ExampleThe gure shows the complete bipartite graph K,, with the colors indicating a partition of its edges into three forests.K, cannot be partitioned into fewer forests, because any forest on its eight vertices has at most seven edges, whilethe overall graph has sixteen edges, more than double the number of edges in a single forest. Therefore, the arboricityof K, is three.

3.2 Arboricity as a measure of densityThe arboricity of a graph is a measure of how dense the graph is: graphs with many edges have high arboricity, andgraphs with high arboricity must have a dense subgraph.In more detail, as any n-vertex forest has at most n-1 edges, the arboricity of a graph with n vertices and m edges isat least dm/(n 1)e . Additionally, the subgraphs of any graph cannot have arboricity larger than the graph itself,or equivalently the arboricity of a graph must be at least the maximum arboricity of any of its subgraphs. Nash-Williams proved that these two facts can be combined to characterize arboricity: if we let nS and mS denote thenumber of vertices and edges, respectively, of any subgraph S of the given graph, then the arboricity of the graphequals maxfdmS/(nS 1)eg:Any planar graph with n vertices has at most 3n 6 edges, from which it follows by Nash-Williams formula thatplanar graphs have arboricity at most three. Schnyder used a special decomposition of a planar graph into threeforests called a Schnyder wood to nd a straight-line embedding of any planar graph into a grid of small area.

3.3 AlgorithmsThe arboricity of a graph can be expressed as a special case of a more general matroid partitioning problem, inwhich one wishes to express a set of elements of a matroid as a union of a small number of independent sets. As aconsequence, the arboricity can be calculated by a polynomial-time algorithm (Gabow & Westermann 1992).

3.4 Related conceptsThe star arboricity of a graph is the size of the minimum forest, each tree of which is a star (tree with at most onenon-leaf node), into which the edges of the graph can be partitioned. If a tree is not a star itself, its star arboricity is

5

6 CHAPTER 3. ARBORICITY

A partition of the complete bipartite graph K4,4 into three forests, showing that it has arboricity three.

two, as can be seen by partitioning the edges into two subsets at odd and even distances from the tree root respectively.Therefore, the star arboricity of any graph is at least equal to the arboricity, and at most equal to twice the arboricity.The linear arboricity of a graph is the size of the minimum linear forest (a forest in which all vertices are incidentto at most two edges) into which the edges of the graph can be partitioned. The linear arboricity of a graph is closelyrelated to its maximum degree and its slope number.The pseudoarboricity of a graph is the minimum number of pseudoforests into which its edges can be partitioned.Equivalently, it is the maximum ratio of edges to vertices in any subgraph of the graph. As with the arboricity, thepseudoarboricity has a matroid structure allowing it to be computed eciently (Gabow & Westermann 1992).The thickness of a graph is the minimum number of planar subgraphs into which its edges can be partitioned. Asany planar graph has arboricity three, the thickness of any graph is at least equal to a third of the arboricity, and atmost equal to the arboricity.The degeneracy of a graph is the maximum, over all induced subgraphs of the graph, of the minimum degree of avertex in the subgraph. The degeneracy of a graph with arboricity a is at least equal to a , and at most equal to 2a1. The coloring number of a graph, also known as its Szekeres-Wilf number (Szekeres & Wilf 1968) is always equalto its degeneracy plus 1 (Jensen & Toft 1995, p. 77f.).The strength of a graph is a fractional value whose integer part gives the maximum number of disjoint spanning treesthat can be drawn in a graph. It is the packing problem that is dual to the covering problem raised by the arboricity.

3.5. REFERENCES 7

The two parameters have been studied together by Tutte and Nash-Williams.The fractional arboricity is a renement of the arboricity, as it is dened for a graphG as maxfmS/(nS 1)jS Gg: In other terms, the arboricity of a graph is the ceiling of the fractional arboricity.The (a,b)-decomposability generalizes the arboricity. A graph is (a; b) -decomposable if its edges can be partitionedinto a + 1 sets, each one of them inducing a forest, except one who induces a graph with maximum degree b . Agraph with arboricity a is (a; 0) -decomposable.

3.5 References Alon, N. (1988). The linear arboricity of graphs. Israel Journal ofMathematics 62 (3): 311325. doi:10.1007/BF02783300.MR 0955135.

Chen, B.; Matsumoto, M.; Wang, J.; Zhang, Z.; Zhang, J. (1994). A short proof of Nash-Williams theoremfor the arboricity of a graph. Graphs and Combinatorics 10 (1): 2728. doi:10.1007/BF01202467. MR1273008.

Erds, P.; Hajnal, A. (1966). On chromatic number of graphs and set-systems. Acta Mathematica Hungarica17 (12): 6199. doi:10.1007/BF02020444. MR 0193025.

Gabow, H. N.; Westermann, H. H. (1992). Forests, frames, and games: Algorithms for matroid sums andapplications. Algorithmica 7 (1): 465497. doi:10.1007/BF01758774. MR 1154585.

Hakimi, S. L.; Mitchem, J.; Schmeichel, E. E. (1996). Star arboricity of graphs. Discrete Mathematics 149:9398. doi:10.1016/0012-365X(94)00313-8. MR 1375101.

Jensen, T. R.; Toft, B. (1995). Graph Coloring Problems. New York: Wiley-Interscience. ISBN 0-471-02865-7. MR 1304254.

C. St. J. A. Nash-Williams (1961). Edge-disjoint spanning trees of nite graphs. Journal of the LondonMathematical Society 36 (1): 445450. doi:10.1112/jlms/s1-36.1.445. MR 0133253.

C. St. J. A. Nash-Williams (1964). Decomposition of nite graphs into forests. Journal of the LondonMathematical Society 39 (1): 12. doi:10.1112/jlms/s1-39.1.12. MR 0161333.

W. Schnyder (1990). Embedding planar graphs on the grid. Proc. 1st ACM/SIAM Symposium on DiscreteAlgorithms (SODA). pp. 138148.

Szekeres, G.; Wilf, H. S. (1968). An inequality for the chromatic number of a graph. Journal of Combina-torial Theory. doi:10.1016/s0021-9800(68)80081-x. MR 0218269.

Tutte, W. T. (1961). On the problem of decomposing a graph into n connected factors. Journal of theLondon Mathematical Society 36 (1): 221230. doi:10.1112/jlms/s1-36.1.221. MR 0140438.

Chapter 4

Aronszajn tree

In set theory, an Aronszajn tree is an uncountable tree with no uncountable branches and no uncountable levels. Forexample, every Suslin tree is an Aronszajn tree. More generally, for a cardinal , a -Aronszajn tree is a tree ofheight such that all levels have size less than and all branches have height less than (so Aronszajn trees are thesame as @1 -Aronszajn trees). They are named for Nachman Aronszajn, who constructed an Aronszajn tree in 1934;his construction was described by Kurepa (1935).A cardinal for which no -Aronszajn trees exist is said to have the tree property. (sometimes the condition that is regular and uncountable is included.)

4.1 Existence of -Aronszajn treesKnigs lemma states that @0 -Aronszajn trees do not exist.The existence of Aronszajn trees (= @1 -Aronszajn trees) was proven by Nachman Aronszajn, and implies that theanalogue of Knigs lemma does not hold for uncountable trees.The existence of @2 -Aronszajn trees is undecidable (assuming a certain large cardinal axiom): more precisely,the continuum hypothesis implies the existence of an @2 -Aronszajn tree, and Mitchell and Silver showed that it isconsistent (relative to the existence of a weakly compact cardinal) that no @2 -Aronszajn trees exist.Jensen proved that V=L implies that there is a -Aronszajn tree (in fact a -Suslin tree) for every innite successorcardinal .Cummings & Foreman (1998) showed (using a large cardinal axiom) that it is consistent that no @n -Aronszajn treesexist for any nite n other than 1.If is weakly compact then no -Aronszajn trees exist. Conversely if is inaccessible and no -Aronszajn trees existthen is weakly compact.

4.2 Special Aronszajn treesAn Aronszajn tree is called special if there is a function f from the tree to the rationals so that f(x)

4.4. SEE ALSO 9

The elements of the tree are certain well-ordered sets of rational numbers with supremum that is rational or . Ifx and y are two of these sets then we dene xy (in the tree order) to mean that x is an initial segment of the orderedset y. For each countable ordinal we write U for the elements of the tree of level , so that the elements of Uare certain sets of rationals with order type . The special Aronszajn tree is the union of the sets U for all countable.We construct U by transnite induction on as follows.

If +1 is a successor then U consists of all extensions of a sequence x in U by a rational greater than supx.

If is a limit then let T be the tree of all points of level less than . For each x in T and for each rationalnumber q greater than sup x, choose a level branch of T containing x with supremum q. Then U consistsof these branches.

The function f(x) = sup x is rational or , and has the property that if x

Chapter 5

Bethe lattice

Cayley tree redirects here. For nite trees with equal-length root-to-leaf paths, see ordered Bell number.A Bethe lattice or Cayley tree (a particular kind of Cayley graph), introduced by Hans Bethe in 1935, is an innite

32101

12

22

2

2 33

3

3333

33

33

A Bethe lattice with coordination number z = 3

connected cycle-free graph where each node is connected to z neighbours, where z is called the coordination number.

10

5.1. RELATION TO CAYLEY GRAPHS 11

It is a rooted tree, with all other nodes arranged in shells around the root node, also called the origin of the lattice.The number of nodes in the kth shell is given by

Nk = z(z 1)k1 for k > 0:

In some situations the denition is modied to specify that the root node has z 1 neighbors.Due to its distinctive topological structure, the statistical mechanics of lattice models on this graph are often exactlysolvable. The solutions are related to the often used Bethe approximation for these systems.

5.1 Relation to Cayley graphsFurther information: Cayley graph

The Bethe lattice where each node is joined to 2n others is essentially the Cayley graph of a free group on n generators.A presentation of a group G by n generators corresponds to a surjective map from the free group on n generatorsto the group G, and at the level of Cayley graphs to a map from the Cayley tree to the Cayley graph. This canalso be interpreted (in algebraic topology) as the universal cover of the Cayley graph, which is not in general simplyconnected.The distinction between a Bethe lattice and a Cayley tree is that the former is innite, while the latter is nite, so thata Bethe lattice has no surface and no root, whereas in Cayley trees the surface is highly non-negligible.[1]

5.2 Lattices in Lie groupsBethe lattices also occur as the discrete group subgroups of certain hyperbolic Lie groups, such as the Fuchsian groups.As such, they are also lattices in the sense of a lattice in a Lie group.

5.3 See also Crystal

5.4 References[1] Ostilli, M. (2012). Cayley Trees and Bethe Lattices, a concise analysis for mathematicians and physicists. Physica A 391:

3417. arXiv:1109.6725v2. Bibcode:2012PhyA..391.3417O. doi:10.1016/j.physa.2012.01.038.

Bethe, H. A. (1935). Statistical theory of superlattices. Proc. Roy. Soc. London Ser A 150: 552575.Bibcode:1935RSPSA.150..552B. doi:10.1098/rspa.1935.0122. Zbl 0012.04501.

Baxter, Rodney J. (1982). Exactly solved models in statistical mechanics. Academic Press. ISBN 0-12-083182-1. Zbl 538.60093.

Chapter 6

Block graph

Not to be confused with block diagram or bar chart.In graph theory, a branch of combinatorial mathematics, a block graph or clique tree[1] is a type of undirected

A block graph

graph in which every biconnected component (block) is a clique.Block graphs are sometimes erroneously called Husimi trees (after Kdi Husimi),[2] but that name more properlyrefers to cactus graphs, graphs in which every nontrivial biconnected component is a cycle.[3]

Block graphs may be characterized as the intersection graphs of the blocks of arbitrary undirected graphs.[4]

6.1 CharacterizationBlock graphs are exactly the graphs for which, for every four vertices u, v, x, and y, the largest two of the threedistances d(u,v) + d(x,y), d(u,x) + d(v,y), and d(u,y) + d(v,x) are always equal.[2][5]

They also have a forbidden graph characterization as the graphs that do not have the diamond graph or a cycle offour or more vertices as an induced subgraph; that is, they are the diamond-free chordal graphs.[5] They are also the

12

6.2. RELATED GRAPH CLASSES 13

Ptolemaic graphs (chordal distance-hereditary graphs) in which every two nodes at distance two from each other areconnected by a unique shortest path,[2] and the chordal graphs in which every two maximal cliques have at most onevertex in common.[2]

A graph G is a block graph if and only if the intersection of every two connected subsets of vertices of G is empty orconnected. Therefore, the connected subsets of vertices in a connected block graph form a convex geometry, a prop-erty that is not true of any graphs that are not block graphs.[6] Because of this property, in a connected block graph,every set of vertices has a unique minimal connected superset, its closure in the convex geometry. The connectedblock graphs are exactly the graphs in which there is a unique induced path connecting every pair of vertices.[1]

6.2 Related graph classesBlock graphs are chordal and distance-hereditary. The distance-hereditary graphs are the graphs in which every twoinduced paths between the same two vertices have the same length, a weakening of the characterization of blockgraphs as having at most one induced path between every two vertices. Because both the chordal graphs and thedistance-hereditary graphs are subclasses of the perfect graphs, block graphs are perfect.Every tree is a block graph. Another class of examples of block graphs is provided by the windmill graphs.Every block graph has boxicity at most two.[7]

Block graphs are examples of pseudo-median graphs: for every three vertices, either there exists a unique vertex thatbelongs to shortest paths between all three vertices, or there exists a unique triangle whose edges lie on these threeshortest paths.[7]

The line graphs of trees are exactly the block graphs in which every cut vertex is incident to at most two blocks, orequivalently the claw-free block graphs. Line graphs of trees have been used to nd graphs with a given number ofedges and vertices in which the largest induced subgraph that is a tree is as small as possible.[8]

6.3 Block graphs of undirected graphsIf G is any undirected graph, the block graph of G, denoted B(G), is the intersection graph of the blocks of G: B(G)has a vertex for every biconnected component of G, and two vertices of B(G) are adjacent if the corresponding twoblocks meet at an articulation vertex. If K1 denotes the graph with one vertex, then B(K1) is dened to be the emptygraph. B(G) is necessarily a block graph: it has one biconnected component for each articulation vertex of G, andeach biconnected component formed in this way must be a clique. Conversely, every block graph is the graph B(G)for some graph G.[4] If G is a tree, then B(G) coincides with the line graph of G.The graph B(B(G)) has one vertex for each articulation vertex of G; two vertices are adjacent in B(B(G)) if theybelong to the same block in G.[4]

6.4 References[1] Vukovi, Kristina (2010), Even-hole-free graphs: A survey, Applicable Analysis and Discrete Mathematics 4 (2): 219

240, doi:10.2298/AADM100812027V.

[2] Howorka, Edward (1979), On metric properties of certain clique graphs, Journal of Combinatorial Theory, Series B 27(1): 6774, doi:10.1016/0095-8956(79)90069-8.

[3] See, e.g., MR 0659742, a 1983 review by Robert E. Jamison of another paper referring to block graphs as Husimi trees;Jamison attributes the mistake to an error in a book by Behzad and Chartrand.

[4] Harary, Frank (1963), A characterization of block-graphs, CanadianMathematical Bulletin 6 (1): 16, doi:10.4153/cmb-1963-001-x.

[5] Bandelt, Hans-Jrgen; Mulder, Henry Martyn (1986), Distance-hereditary graphs, Journal of Combinatorial Theory,Series B 41 (2): 182208, doi:10.1016/0095-8956(86)90043-2.

[6] Edelman, Paul H.; Jamison, Robert E. (1985), The theory of convex geometries, Geometriae Dedicata 19 (3): 247270,doi:10.1007/BF00149365.

14 CHAPTER 6. BLOCK GRAPH

[7] Block graphs, Information System on Graph Class Inclusions.

[8] Erds, Paul; Saks, Michael; Ss, Vera T. (1986), Maximum induced trees in graphs, Journal of Combinatorial Theory,Series B 41 (1): 6179, doi:10.1016/0095-8956(86)90028-6.

Chapter 7

Borvkas algorithm

An animation, describing Boruvkas (Sollins) algorithm, for nding a minimum spanning tree in a graph - An example on the runtimeof the algorithm

Borvkas algorithm is an algorithm for nding a minimum spanning tree in a graph for which all edge weights aredistinct.It was rst published in 1926 by Otakar Borvka as a method of constructing an ecient electricity network forMoravia.[1][2][3] The algorithmwas rediscovered byChoquet in 1938;[4] again by Florek, ukasiewicz, Perkal, Steinhaus,and Zubrzycki[5] in 1951; and again by Sollin [6] in 1965. Because Sollin was the only computer scientist in this listliving in an English speaking country, this algorithm is frequently called Sollins algorithm, especially in the parallelcomputing literature.The algorithm begins by rst examining each vertex and adding the cheapest edge from that vertex to another in thegraph, without regard to already added edges, and continues joining these groupings in a like manner until a treespanning all vertices is completed.

7.1 PseudocodeDesignating each vertex or set of connected vertices a "component", pseudocode for Borvkas algorithm is:

1. Input: A connected graph G whose edges have distinct weights

15

16 CHAPTER 7. BORVKAS ALGORITHM

2. Initialize a forest T to be a set of one-vertex trees, one for each vertex of the graph.

3. While T has more than one component:

4. For each component C of T:

5. Begin with an empty set of edges S

6. For each vertex v in C:

7. Find the cheapest edge from v to a vertex outside of C, and add it to S

8. Add the cheapest edge in S to T

9. Output: T is the minimum spanning tree of G.

As in Kruskals algorithm, tracking components of T can be done eciently using a disjoint-set data structure. Ingraphs where edges have identical weights, edges with equal weights can be ordered based on the lexicographic orderof their endpoints.

7.2 ComplexityBorvkas algorithm can be shown to take O(log V) iterations of the outer loop until it terminates, and therefore torun in time O(E log V), where E is the number of edges, and V is the number of vertices in G. In planar graphs, andmore generally in families of graphs closed under graph minor operations, it can be made to run in linear time, byremoving all but the cheapest edge between each pair of components after each stage of the algorithm.[7]

7.3 Example

7.4 Other algorithmsOther algorithms for this problem include Prims algorithm and Kruskals algorithm. Fast parallel algorithms can beobtained by combining Prims algorithm with Borvkas.[8]

A faster randomizedminimum spanning tree algorithm based in part on Borvkas algorithm due to Karger, Klein, andTarjan runs in expected O(E) time.[9] The best known (deterministic) minimum spanning tree algorithm by BernardChazelle is also based in part on Borvkas and runs in O(E (E,V)) time, where is the inverse of the Ackermannfunction.[10] These randomized and deterministic algorithms combine steps of Borvkas algorithm, reducing thenumber of components that remain to be connected, with steps of a dierent type that reduce the number of edgesbetween pairs of components.

7.5 Notes[1] Borvka, Otakar (1926). O jistm problmu minimlnm (About a certain minimal problem)". Prce mor. prodovd.

spol. v Brn III (in Czech and German summary) 3: 3758.

[2] Borvka, Otakar (1926). Pspvek k een otzky ekonomick stavby elektrovodnch st (Contribution to the solutionof a problem of economical construction of electrical networks)". Elektronick Obzor (in Czech) 15: 153154.

[3] Neetil, Jaroslav; Milkov, Eva; Neetilov, Helena (2001). Otakar Borvka on minimum spanning tree problem:translation of both the 1926 papers, comments, history. Discrete Mathematics 233 (13): 336. doi:10.1016/S0012-365X(00)00224-7. MR 1825599.

[4] Choquet, Gustave (1938). "tude de certains rseaux de routes. Comptes-rendus de lAcadmie des Sciences (in French)206: 310313.

[5] Florek, Kazimierz (1951). Sur la liaison et la division des points d'un ensemble ni. Colloquium Mathematicum 2 (1951)(in French): 282285.

7.5. NOTES 17

[6] Sollin, M. (1965). Le trac de canalisation. Programming, Games, and Transportation Networks (in French).

[7] Eppstein, David (1999). Spanning trees and spanners. In Sack, J.-R.; Urrutia, J. Handbook of Computational Geometry.Elsevier. pp. 425461.; Mare, Martin (2004). Two linear time algorithms for MST on minor closed graph classes(PDF). Archivum mathematicum 40 (3): 315320..

[8] Bader, David A.; Cong, Guojing (2006). Fast shared-memory algorithms for computing the minimum spanning forest ofsparse graphs. Journal of Parallel and Distributed Computing 66 (11): 13661378. doi:10.1016/j.jpdc.2006.06.001.

[9] Karger, David R.; Klein, Philip N.; Tarjan, Robert E. (1995). A randomized linear-time algorithm to nd minimumspanning trees. Journal of the ACM 42 (2): 321328. doi:10.1145/201019.201022.

[10] Chazelle, Bernard (2000). Aminimum spanning tree algorithm with inverse-Ackermann type complexity (PDF). J. ACM47 (6): 10281047. doi:10.1145/355541.355562.

Chapter 8

Branch-decomposition

a b c

d e f

g h i

ab be

bc

cf

ef

fihieh

gh

dg

de

ad

Branch decomposition of a grid graph, showing an e-separation. The separation, the decomposition, and the graph all have widththree.

In graph theory, a branch-decomposition of an undirected graph G is a hierarchical clustering of the edges of G,represented by an unrooted binary tree T with the edges of G as its leaves. Removing any edge from T partitions theedges of G into two subgraphs, and the width of the decomposition is the maximum number of shared vertices of anypair of subgraphs formed in this way. The branchwidth of G is the minimum width of any branch-decompositionof G.Branchwidth is closely related to tree-width: for all graphs, both of these numbers are within a constant factor of eachother, and both quantities may be characterized by forbiddenminors. And as with treewidth, many graph optimizationproblems may be solved eciently for graphs of small branchwidth. However, unlike treewidth, the branchwidth ofplanar graphs may be computed exactly, in polynomial time. Branch-decompositions and branchwidth may also begeneralized from graphs to matroids.

8.1 DenitionsAn unrooted binary tree is a connected undirected graph with no cycles in which each non-leaf node has exactlythree neighbors. A branch-decomposition may be represented by an unrooted binary tree T, together with a bijection

18

8.2. RELATION TO TREEWIDTH 19

between the leaves of T and the edges of the given graph G = (V,E). If e is any edge of the tree T, then removing efrom T partitions it into two subtrees T1 and T2. This partition of T into subtrees induces a partition of the edgesassociated with the leaves of T into two subgraphs G1 and G2 of G. This partition of G into two subgraphs is calledan e-separation.The width of an e-separation is the number of vertices of G that are incident both to an edge of E1 and to an edgeof E2; that is, it is the number of verti