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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 Trees and block graphs
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.
Flavia Bonomo (DC–FCEN–UBA) SATG 2016 10 / 52
Intersection graphs Trees and block graphs
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
Flavia Bonomo (DC–FCEN–UBA) SATG 2016 12 / 52
Intersection graphs Chordal graphs
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.
Flavia Bonomo (DC–FCEN–UBA) SATG 2016 13 / 52
Intersection graphs Chordal graphs
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
Flavia Bonomo (DC–FCEN–UBA) SATG 2016 14 / 52
Intersection graphs Chordal graphs
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].
Flavia Bonomo (DC–FCEN–UBA) SATG 2016 15 / 52
Intersection graphs Chordal graphs
Perfect elimination ordering
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
Flavia Bonomo (DC–FCEN–UBA) SATG 2016 16 / 52
Intersection graphs Chordal graphs
Perfect elimination ordering
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
Flavia Bonomo (DC–FCEN–UBA) SATG 2016 17 / 52
Intersection graphs Chordal graphs
Algorithmic problems in chordal graphs
How can we solve maximum clique, maximum stable set, minimumcoloring and minimum clique-cover on chordal graphs?
Flavia Bonomo (DC–FCEN–UBA) SATG 2016 18 / 52
Intersection graphs Chordal graphs
Algorithmic problems in 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 ])
Flavia Bonomo (DC–FCEN–UBA) SATG 2016 19 / 52
Intersection graphs Chordal graphs
Algorithmic problems in chordal graphs
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?
Flavia Bonomo (DC–FCEN–UBA) SATG 2016 20 / 52
Intersection graphs Chordal graphs
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.
Flavia Bonomo (DC–FCEN–UBA) SATG 2016 21 / 52
Intersection graphs Chordal graphs
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
Intersection graphs Chordal graphs
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
Flavia Bonomo (DC–FCEN–UBA) SATG 2016 23 / 52
Intersection graphs Chordal graphs
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 ).
Flavia Bonomo (DC–FCEN–UBA) SATG 2016 24 / 52
Intersection graphs Chordal graphs
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?
Flavia Bonomo (DC–FCEN–UBA) SATG 2016 25 / 52
Intersection graphs Chordal graphs
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.
Flavia Bonomo (DC–FCEN–UBA) SATG 2016 26 / 52
Intersection graphs Chordal graphs
Chordal graphs as intersection graphs
Theorem (Gavril, 1974)
A graph is chordal iff it is the intersection graph of some subtrees of a tree.
Flavia Bonomo (DC–FCEN–UBA) SATG 2016 27 / 52