Algorithmic Graph Theory

Treewidth

Outline • Tree decomposition - treewidth (tw(G))

• Path decomposition - pathwidth (pw(G))

• partial k-trees

• chordal graphs

• interval graphs

• Connections with tw(G) and pw(G) – chordal graphs and interval graphs

– graph searching

Example

Example

Example

Example

tw(G) = treewidth of G

ω(G) = clique number of G

Example

tw(G) = 2

width(X, T) = 2

Example

Example

pw(G) = 2 (???)

width(X, P) = 3

computing tw(G) and pw(G)

computing tw(G) and pw(G)

k-trees

partial k-trees and tw(G)

Theorem: G is a partial k-tree tw(G) ≤ k

if G is a k-tree then tw(G) ≤ k

partial k-tree k-tree and tw(H) ≤ tw(G) for H G

Induction on |V|: pick v, N(v) = K, K Xi , Xj = K {v}, Xi<-> Xj

if tw(G) ≤ k then G is a partial k-tree

Pick a leaf node Xleaf, Xj <-> Xleaf , such that Xleaf – Xj = {v}

| Xleaf | ≤ k+1: |N(v)| ≤ k and N(v) Xj (since v Xi)

By induction G[Xj] is a clique: N(v) induces a clique

http://www.graphclasses.org/

• tw(G) = 1 G forest,

• tw(G) = 2 G series parallel,

• tw(G) = 3 G partial 3-tree,

• ...

Hasse diagram of subclasses

chordal graphs

chordal graphs

chordal graphs

• Pls., prove Theorem 5.2

simplicial vertices

simplicial vertices

chordal graphs

Pls., prove Lemma

Triangulations

• Triangulation: Any graph G can be turned into a chordal graph by adding edges

• Use the perfect elimination ordering (elimination game): – Choose any vertex x to start with, and add the necessary edges so that

the neighbors of x become a clique.

– Remove x from the modied graph, and continue this process until all vertices are processed

• Filled graph: the resulting graph

• Triangulations can also be computed in other ways: – for example by making every minimal separator into a clique by adding

edges

• Computing a triangulation with the min number of edges: – computing a "nice" ordering β among all of them

– Unfortunately, this is an NP-hard problem

Triangulations

Order: < a, b, c, d, e, f >

added edges: {} , {cf}, {df}, {}, {}, {}

Triangulations

Order: < a, b, c, d, e, f >

added edges: {} , {cf}, {df}, {}, {}, {}

Order: < a, c, d, b, e, f >

added edges: {} , {bd}, {df}, {}, {}, {}

Triangulations

Order: < a, b, c, d, e, f >

added edges: {} , {cf}, {df}, {}, {}, {}

Order: < a, c, d, b, e, f >

added edges: {} , {bd}, {df}, {}, {}, {}

chordal graphs and maximal cliques

chordal graphs and maximal cliques

chordal graphs and maximal cliques

chordal graphs and clique trees

Clique tree T (iii): There exists a tree whose vertex set is the set of maximal cliques of G (each T[Kv] is connected)

• If G is a chordal graph, then any clique tree of G is also a tree decomposition of G (the opposite is not always true)

• Useful structure to express the information on maximal cliques and minimal separators of a chordal graph.

1

2

3 4

clique graph

• (Ci,Cj) E Ci Cj

• w(e)= | Ci Cj |

clique graph and clique tree

• Note that the clique graph of G is unique, whereas G may have several different clique trees

clique graph and clique tree

• number of minimal separators < |V|

connection with Tree Decomposition

clique trees on interval graphs

clique trees on interval graphs

clique trees on interval graphs

clique trees on interval graphs

• Its clique tree = tree decomposition = path decomposition

clique trees on interval graphs • clique trees are paths, clique trees are path decompositions

• pathwidth = treewidth for interval graphs

• pw(G) = |largest clique| – 1 when G is an interval graph

• For general graph G:

Example

pw(G) = 2 (???)

width(X, P) = 3

Example

pw(G) = 2 (???)

Example

pw(G) = 2 (???)

Finding an interval completion where the size of the largest clique is as small as possible is an NP-hard problem. Even for chordal graphs it is an open question whether or not such an interval completion can be found in polynomial time.

Outline • Tree decomposition - treewidth (tw(G))

• Path decomposition - pathwidth (pw(G))

• partial k-trees

• chordal graphs

• interval graphs

• Connections with tw(G) and pw(G) – chordal graphs and interval graphs

– graph searching

graph searching