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Monotone Drawings of Graphs

Thanks to Peter Eades

www.dia.uniroma3.it/~compunet

Patrizio Angelini, Enrico Colasante,

Giuseppe Di Battista, Fabrizio Frati, and Maurizio Patrignani

Roma Tre University, Italy

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Konstanz Univ.

Bellavista Hotel

direction of monotonicity

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Konstanz Univ.

Petershof hotel

no direction of monotonicity

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monotone paths• monotone path with respect to a half-line l

– each segment of the path has a positive projection onto l• monotone path

– there exists an l such that the path is monotone with respect to

l

lp1

p2

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monotone drawings of graphs• monotone (straight-line) drawing of a

graph G– each pair of vertices of G are joined by a

monotone path• monotonicity does not imply planarity

l

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overview of the talk

• properties of monotone drawings

• monotone drawings of trees

• monotone drawings of graphs

• planar monotone drawings of biconnected graphs

• conclusions and open problems

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properties of monotone drawings• each pair of adjacent

edges forms a monotone path

• any subpath of a monotone path is monotone

• affine transformations preserve monotonicity

• each monotone path is planar– a monotone drawing

of a tree is planar

notmonotone

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convex drawings of trees• a convex drawing of a tree is such that replacing

edges leading to leaves with half-lines yields a partition of the plane into convex unbounded regions [Carlson, Eppstein, GD’06]

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strictly convex drawings of trees• a strictly convex drawing of a tree T is such that:

– it is a convex drawing of T– each set of parallel edges of T forms a collinear path

• every strictly convex drawing of a tree is monotone

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slope-disjoint drawing of trees• slope-disjoint drawing of a tree T

– each subtree rooted at vertex v uses an interval of slopes (v, v) where v – v <

– if u is the parent of v you have v < u < s(u,v) < u < v

– if v and w are siblings (v, v) (w, w) = v

u

v

u

v

w

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slope-disjoint drawing of trees• any slope-disjoint drawing of a tree is

monotone• we propose two algorithms:

– BFS-based algorithm• constructs a monotone drawing of a tree on a grid of area

O(n1.6) O(n1.6) – DFS-based algorithm

• constructs a monotone drawing of a tree on a grid of area O(n2) O(n)

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monotone drawings of graphs

• any graph admits a monotone drawing– consider a spanning tree T of the input graph– produce a monotone drawing of T– add the remaining edges

• the produced drawings may have crossings even if the input graph is planar

is it possible to have planar monotone drawings of planar graphs?

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biconnected graphs• a cut-vertex is a vertex such

that its removal produces a disconnected graph

• a biconnected graph does not have cut-vertices

• a separation pair of a biconnected graph is a pair of vertices whose removal produces a disconnected graph

• a split pair is either a separation pair or a pair of adjacent vertices

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SPQR-tree

u

v

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SPQR-tree

Q

u

v

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SPQR-tree

Q

u

v

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SPQR-tree

Q

Su

v

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SPQR-tree

Q

S

Q

u

v

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SPQR-tree

Q

S

Q P

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SPQR-tree

Q

Q

Q

Q Q

S

P

S

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SPQR-tree

Q

Q

Q

Q Q

S

P

Q Q Q Q Q

R

S

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SPQR-tree

Q

Q

Q

Q Q

S

P

Q Q Q Q Q

R

S

• each internal node of the tree is associated with a skeleton representing its configuration

• the graph represented by node into its parent is called the pertinent of

u

v

u

v

skeleton of S

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convex drawings are monotone• graphs admitting strictly convex drawings

[Chiba, Nishizeki, 88]– are biconnected – have an embedding such that each split pair u,v

• is incident to the outer face• all its maximal split components, with the possible exception

of edge (u,v), have at least one edge on the outer face

• any strictly convex drawing of a graph is monotone [Arkin, Connelly, Mitchell, SoCG ‘89]

• with similar techniques we show that– any non-strictly convex drawing of a graph such that

each set of parallel edges forms a collinear path is monotone

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strategy for biconnected graphs• we apply an inductive algorithm

to the nodes of the SPQR-tree• a node of the tree is

associated with a quadrilateral shape called boomerang of and denoted by boom()

• the pertinent of uses a “restricted” range of slopes

• the boomerangs of the children of are arranged into boom()

N

S

EW

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strategy for biconnected graphs

• invariants on boom()– symmetric with respect to

the line through W and E

– angle + 2 < /2

N

S

EW

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properties of the drawings• the inductive algorithm

constructs a drawing of pert() such that is monotone– is contained into boom()

• with the possible exception of the edge joining the poles of

– any vertex w of pert() belongs to a path that is a composition of

• a path toward N which is monotone with respect to dN and uses slopes in (dN-, dN+)

• a path toward S monotone w.r.t. dS and using slopes in (dS-, dS+)

N

E

S

W

dS

dN

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base case: Q-node

• if is a Q-node draw the corresponding edge as a segment from N to S

N

S

EW

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if is an S-node

• find the intersection between the two bisectors of angles

• arrange the boomerangs of the children as in the figure

• recur on the children using ’ + 2’ < /2

N

E

S

W

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if is a P-node

• split boom() into a suitable number of slices

• compute and for each slice

• recur on the children

N

E

S

W

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if is an R-node• remove the south pole and obtain

graph G’• observe that G’ admits a convex

drawing into any convex polygon• draw G’ into a suitable convex

polygon in the upper portion of boom()

• squeeze the drawing towards the N-E border of boom() in order to make sure that the drawing uses a restricted interval of slopes

• compute and for each child• recur on the children

N

E

S

W

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admitting planar and monotone drawings

admitting strictly convex drawings

planarbiconnected

planartriconnectedtrees

planar simply connected

?

?

??

?

?

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conclusions and open problems• address simply connected graphs• determine tight bounds on the area

requirements for grid drawings of trees• devise algorithms to construct monotone

drawings of non-planar graphs on a grid of polynomial size

• construct monotone drawings of biconnected graphs in polynomial area

• explore strongly monotone drawings, where each pair of vertices u,v has a joining path that is monotone with respect to the line from u to v

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change my embedding if you want me monotone

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thank you

thank you