Clique and neighborhood characterizations of strongly ...

Clique and neighborhood characterizationsof strongly chordal graphs

Pablo De Caria 1 and Terry McKee 2

1 CONICET/ Departamento de Matematica, Universidad Nacional de La Plata2 Department of Mathematics and Statistics, Wright State University, Dayton,

Ohio

6th Latin American Workshop on Cliques in Graphs,Pirenopolis, November 2014

Clique and neighborhood characterizations of strongly chordal graphs

Chordal graphs

A chord of a cycle is an edge joining two nonconsecutive verticesof the cycle.

A graph G is chordal if every cycle of G with length at least fourhas a chord.

Strongly chordal graphs

A strong chord in a cycle C of even length is a chord joining twovertices of C at odd distance.

A chordal graph G is strongly chordal if every cycle of G withlength even and at least 6 has a strong chord.


Chordal graphs







Chordal graphs







The n-sun, n ≥ 3, is the graph with vertices v1, ..., vn,w1, ...,wn

such that {v1, ..., vn} is a clique, N(wn) = {v1, vn} andN(wi ) = {vi , vi+1}, for 1 ≤ i ≤ n − 1.

A graph is strongly chordal if and only if it is chordal and sun-free.

Another characterization

A graph is strongly chordal if and only it is a hereditary duallychordal graph.














Simple verticesv is a simple vertex of G if there exists an order w1, ...,wn for theneighbors of v such that N[v ] ⊆ N[w1] ⊆ ... ⊆ N[wn].

An ordering v1v2...vn of all the vertices of G is simple if vi issimple in G [{vi , ..., vn}] for every 1 ≤ i ≤ n.

One more characterizationG is strongly chordal if and only if G has a simple eliminationordering.


Simple verticesv is a simple vertex of G if there exists an order w1, ...,wn for theneighbors of v such that N[v ] ⊆ N[w1] ⊆ ... ⊆ N[wn].

An ordering v1v2...vn of all the vertices of G is simple if vi issimple in G [{vi , ..., vn}] for every 1 ≤ i ≤ n.

One more characterizationG is strongly chordal if and only if G has a simple eliminationordering.


Our goals

I Use clique related parameters to obtain characterizations forchordal graphs.

I Add more conditions to those characterizations to obtain newcharacterizations of strongly chordal graphs.

I Show that a very similar approach is possible using closedneighborhoods instead of cliques.


Clique strengthThe clique strength of S ⊆ V (G ), or cstrG (S), is the number ofcliques of G that contain S .

A subgraph H of G is a strength-k subgraph if cstrG (V (H)) ≥ k.

The definition of chordal graphs in terms of clique strength:G is chordal if and only if every cycle of strength-1 edges eitherhas a strength-1 chord or is a strength-1 triangle.

The associated characterization of strongly chordal graphs[McKee, 1999]G is strongly chordal if and only if, for every k ≥ 1, every cycle ofcstrG − k edges either has a cstrG − k chord or is a cstrG − ktriangle.












Disk strengthThe disk strength of S ⊆ V (G ), or dstrG (S), is the number ofdifferent closed neighborhoods in G that contain S .

A subgraph H of G is a disk strength-k subgraph, or dstr − ksubgraph, if dstrG (V (H)) ≥ k.

A graph G is chordal if and only if every cycle of dstr − 1 edgeseither contains a dstr − 1 chord or is a dstr − 1 triangle.










Disk strength characterization of strongly chordal graphs

G is strongly chordal if and only if, for every k ≥ 1, every cycle ofdstr − k edges either has a dstr − k chord or is a dstr − k triangle.

Structure of the proof

It is by induction.

Find a simple vertex v .

Given any cycle C , the cases v ∈ C and v 6∈ C are considered.

In some cases, the conclusion can be directly obtained. In others, itis obtained by considering the strongly chordal graph G − v .


Disk strength characterization of strongly chordal graphs

G is strongly chordal if and only if, for every k ≥ 1, every cycle ofdstr − k edges either has a dstr − k chord or is a dstr − k triangle.

Structure of the proof

It is by induction.

Find a simple vertex v .

Given any cycle C , the cases v ∈ C and v 6∈ C are considered.

In some cases, the conclusion can be directly obtained. In others, itis obtained by considering the strongly chordal graph G − v .


If G is not strongly chordal, then G has a chordless cycle of lengthat least four or an n-sun, n ≥ 3.

In the first case, C is a cycle of dstr − 1 edges without dstr − 1chords that is not a dstr − 1 triangle.

In the second case, the length n cycle whose vertices form decentral clique is a cycle of dstr − (n + 1) edges that has nodstr − (n + 1) chord and is not a dstr − (n + 1) triangle.


If G is not strongly chordal, then G has a chordless cycle of lengthat least four or an n-sun, n ≥ 3.

In the first case, C is a cycle of dstr − 1 edges without dstr − 1chords that is not a dstr − 1 triangle.

In the second case, the length n cycle whose vertices form decentral clique is a cycle of dstr − (n + 1) edges that has nodstr − (n + 1) chord and is not a dstr − (n + 1) triangle.


Euler type characteristics

Ci (G ) : Subsets of cardinality i contained in at least one clique ofG .

Clique characteristic:

X (G ) = c1(G )− c2(G ) + c3(G )− ... =∞∑i=1

(−1)i+1ci (G )

Example

If G is the 3-sun, then c1(G ) = 6, c2(G ) = 9, c3(G ) = 4 andci (G ) = 0 for i ≥ 4.

Thus X (G ) = 6− 9 + 4 = 1.

Characterization [McKee, 1999]

A graph is chordal if and only if every induced subgraph H satisfiesX (H) = Comp(H).





X (G ) = c1(G )− c2(G ) + c3(G )− ... =∞∑i=1

(−1)i+1ci (G )

Example


Thus X (G ) = 6− 9 + 4 = 1.







X (G ) = c1(G )− c2(G ) + c3(G )− ... =∞∑i=1

(−1)i+1ci (G )

Example


Thus X (G ) = 6− 9 + 4 = 1.




The same with closed neighborhoodsdi (G ) : Subsets of cardinality i contained in at least one closedneighborhood.

The disk characteristic:

X (G ) = d1(G )− d2(G ) + d3(G )− ... =∞∑i=1

(−1)i+1di (G )

ExampleIf G is the 3-sun, then d1(G ) = 6, d2(G ) = 15, d3(G ) = 19,d4(G ) = 12, d5(G ) = 3 and di (G ) = 0 whenever i ≥ 6.

Thus X (G ) = 6− 15 + 19− 12 + 3 = 1




X (G ) = d1(G )− d2(G ) + d3(G )− ... =∞∑i=1

(−1)i+1di (G )


Thus X (G ) = 6− 15 + 19− 12 + 3 = 1




X (G ) = d1(G )− d2(G ) + d3(G )− ... =∞∑i=1

(−1)i+1di (G )


Thus X (G ) = 6− 15 + 19− 12 + 3 = 1

CharacterizationA graph is chordal if and only if every induced subgraph H satisfiesX (H) = comp(H).


The condition is necessary

X (C4) = 4− 6 + 4− 0 + ... = 2

For n > 4,

X (Cn) = n − 2n + n − 0 + ... = 0


A generalization

c(k)i (G ): cardinality i subsets contained in at least k cliques of G .

X (k)(G ) = c(k)1 (G )− c

(k)2 (G ) + c

(k)3 (G )− ... =

∞∑i=1

(−1)i+1c(k)i (G ).

G (k): graph formed by the cstr − k vertices and edges of G .

Example

If G is the 3-sun and k = 2, then c(2)1 (G ) = 3, c

(2)2 (G ) = 3 and

c(2)i (G ) = 0 for i ≥ 4

X (2)(G ) = 3− 3 = 0 and G (2) = K3.


A generalization


X (k)(G ) = c(k)1 (G )− c

(k)2 (G ) + c

(k)3 (G )− ... =

∞∑i=1

(−1)i+1c(k)i (G ).


Example


(2)2 (G ) = 3 and

c(2)i (G ) = 0 for i ≥ 4

X (2)(G ) = 3− 3 = 0 and G (2) = K3.


A generalization


X (k)(G ) = c(k)1 (G )− c

(k)2 (G ) + c

(k)3 (G )− ... =

∞∑i=1

(−1)i+1c(k)i (G ).


Example


(2)2 (G ) = 3 and

c(2)i (G ) = 0 for i ≥ 4

X (2)(G ) = 3− 3 = 0 and G (2) = K3.

CharacterizationA graph G is strongly chordal if and only if, for every k ≥ 1, everyinduced subgraph H satisfies X (k)(H) = Comp(H(k)).


The neighborhood version

d(k)i (G ): cardinality i subsets contained in at least k closed

neighborhoods of G .

X (k)(G ) = d(k)1 (G )−d

(k)2 (G )+d

(k)3 (G )− ... =

∞∑i=1

(−1)i+1d(k)i (G ).

G (k): graph formed by the dstr − k vertices and edges of G .


Idea of the proofLet A1, ...,Am be a partition of V (G ) into set of vertices withequal closed neighborhood.

Let G ′ be the graph obtained by adding vertices u1, ..., um suchthat NG ′(ui ) = Ai for 1 ≤ i ≤ m. And let G = (G ′)2.





X (k)(G ) = d(k)1 (G )−d

(k)2 (G )+d

(k)3 (G )− ... =

∞∑i=1

(−1)i+1d(k)i (G ).









X (k)(G ) = d(k)1 (G )−d

(k)2 (G )+d

(k)3 (G )− ... =

∞∑i=1

(−1)i+1d(k)i (G ).






Idea of the proof

G ′ is strongly chordal, so G is also strongly chordal.

The cliques of G are the closed neighborhoods in G ′ of the verticesin V (G ).

Do the same for every induced subgraph H.

For the converse, consider the cycles and suns as usual.

Further reading

Pablo De Caria and Terry McKee, Maxclique and unit diskcharacterizations of strongly chordal graphs, DiscussionesMathematicae Graph Theory 34 (2014) 593-602.


Idea of the proof





Further reading



Idea of the proof





Further reading



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