Seminario Avanzado de Teoría de Grafos

Intersection graphs Trees and block graphs

How to generalize trees? Block graphs

The property of cutpoints in trees is equivalent to every biconnectedcomponent (block) is a vertex or an edge.

Block graph: every biconnected component (block) is a clique.

Forbidden induced subgraphs: Cn, n ≥ 4, diamond.

leaves ! end blocks

Many algorithms for trees extend to block graphs.

Diamond-free ⇔ every edge belongs to one maximal clique.

diamond

Flavia Bonomo (DC–FCEN–UBA) SATG 2016 9 / 52


Intersection graphs

An intersection graph of a (finite) family of sets F has one vertex for eachmember F ∈ F and two vertices F ,F ′ are adjacent iff F ∩ F ′ 6= ∅.

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Is every graph an intersection graph?

Intersection graphs of specific objects can have interesting properties.



Intersection model for block graphs?

Line graph of a graph: intersection of its edges.

G L(G) G L(G)

Line graphs of trees?

Line graphs of trees ( block graphsFlavia Bonomo (DC–FCEN–UBA) SATG 2016 11 / 52

Intersection graphs Chordal graphs

More general: chordal graphs

A graph is chordal if it contains no induced Cn, n ≥ 4, that is, if everycycle of length at least 4 has a chord.

Also called triangulated or rigid circuit.

[recommended lecture] Blair and Peyton, An introduction to chordalgraphs and clique trees



Minimal vertex separators

A subset S ⊂ V is a separator if two vertices that where in the sameconnected component of G are in different connected components ofG \ S .

If a and b are two vertices separated by S , it is called ab-separator.

S is a minimal separator (resp. ab-separator) if no proper subset of Sis a separator (resp. ab-separator).

S is a minimal vertex separator if it is a minimal ab-separator forsome pair of vertices ab.

A minimal vertex separator is not necessarily a minimal separator.



Minimal vertex separators

Theorem (Dirac, 1961)

A graph is chordal if and only if every minimal vertex separator is complete.

⇐) Let v0v1v2...vk , k ≥ 3, a cycle and consider v0v2. Either v0v2 is a chord orthere is a v0v2-separator, that should contain v1 and some vi , 3 ≤ i ≤ k. Since itis complete, v1vi is a chord.

⇒) Let S be a minimal ab-separator andsuppose x , y in S non adjacent. Let A,B be the connected components ofG \ S containing a and b, resp. Since Sis minimal, both x and y have neighborsin A and B. Let PA and PB be pathsbetween x and y with interior in A andB, resp., of minimum length. Then thecycle xPAyPBx has no chords, acontradiction. 2



Perfect elimination ordering

A vertex v is simplicial if N[v ] induces a complete subgraph on G .

An ordering v1, v2, . . . , vn of the vertices of a graph G is a perfectelimination ordering if, for every 2 ≤ i ≤ n − 2 vi is simplicial inG [vi , vi+1, . . . , vn].




Theorem (Dirac, 1961)

Every chordal graph has a simplicial vertex. If it is not complete, then ithas two non-adjacent simplicial vertices.

Proof. Complete graph or n = 2, trivial. Otherwise, induction using minimalvertex separators. 2




Theorem (Fulkerson and Gross, 1965)

A graph is chordal if and only if it has a PEO.

Proof. (⇒) By induction. (⇐) Consider a cycle and the vertex with smallestlabel. 2



Algorithmic problems in chordal graphs

How can we solve maximum clique, maximum stable set, minimumcoloring and minimum clique-cover on chordal graphs?




Let v be a simplicial vertex:

either v belongs to the maximum clique or not... but v belongs to just onemaximal clique! so...

ω(G ) = max{|N[v ]|, ω(G − v)}

Note: there is a linear number of maximal cliques!

a maximum stable set either contains v or contains one of its neighbors w ,but since N[w ] ⊇ N[v ], we can replace it by v , so...

α(G ) = 1 + α(G − N[v ])




Let v be a simplicial vertex:

We can extend an optimum coloring of G − v to G without adding colorsunless χ(G − v) < d(v). But in that case we add one new color and, asN[v ] is a clique, it is optimum. So...

χ(G ) = max{|N[v ]|, χ(G − v)}

We should cover v and it belongs to just one maximal clique, so we useN[v ] and continue...

τ(G ) = 1 + τ(G − N[v ])

Do these recursions recall something?



Chordal graphs are perfect

For every chordal graph G , ω(G ) = χ(G ) and α(G ) = τ(G ), and it holdsfor the induced subgraphs because the class is hereditary.



Minimum dominating set?

We can always avoid using a simplicial vertex (unless G is a clique).But... the problem is NP-complete.

Given a SAT formula on n variables and k clauses, is there a dominatingset in the graph of cardinality n?Flavia Bonomo (DC–FCEN–UBA) SATG 2016 22 / 52


Split graphs

The minimum dominating set problem is NP-complete even in splitgraphs, a subclass of chordal graphs.

A graph G is split if V (G ) can be partitioned into a clique and astable set.

Split = {C4,C5, 2K2}-free



Clique trees

A clique tree of a graph G is a tree T (G ) whose vertices are the maximalcliques of G and such that, for each vertex v ∈ G the maximal cliques ofG containing v form a connected subgraph of T (G ).



Clique trees

Theorem (Gavril/Buneman/Walter, 1974)

A graph is chordal iff it admits a clique tree.

How can we recover minimal vertex separators from a clique tree?



Finding a clique tree

Theorem (Bernstein and Goodman, 1981)

Every clique tree of a chordal graph G is a maximum weight spanning treeof the 2-weighted clique graph of G (Kw

2 (G )), and conversely.



Chordal graphs as intersection graphs

Theorem (Gavril, 1974)

A graph is chordal iff it is the intersection graph of some subtrees of a tree.


Seminario Avanzado de Teoría de Grafos

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