1 Treewidth, partial k-tree and chordal graphs Delpensum INF 334 Institutt fo informatikk Pinar Heggernes Speaker: 李維陞

2 Outline Polynomial time algorithm for maximum independent set. Graph class define symbols Tree decomposition Treewidth Partial k-tree Chordal graph Clique tree Interval graphs

3 Problem Many graph problems that are NP-hard on general graphs. But if these graphs are trees, then we can solve these in polynomial time. Or these general graph problem can be categorized as some kind of graph, such as chodal graph or interval graph. General graphs Some kind of Restricted graph Design technique mapping Polynomial time NP-hard

4 Classification(1) A graph is called a  graph if it satisfies property . For example, a graph having no induced cycle of length greater than 3 is called a chordal graph. Many problems in graphs become polynomial-time solvable if the graphs considered satisfy some nice property .

5 Classification(2) Tree Just about all graph problems become linear-time sovable on trees. Planar graphs The maximum cut problem, NAE-3SAT become polynomial time solvable on plannar graph. Chodal graphs Maximum independent set problem There are almost 200 graph classes described in the book “Graph Classes, A Survey” by Brandstädt, Le, and Spinrad, published in 1999.

6 Perfect GraphsCocomparabilityComparabilityCircle GraphsWeakly ChordalChordal Distance-Hereditary PermutationCographsSplitStrongly ChordalTreesIntervalCircular-Arc Some Graph Classes

7 definition(1) G=(V,E) N(v): the neighbors of a vertex v, v ∈ V. ω(G): The number of graph vertices in the largest clique of G. κ(G): the size of a smallest possilbe clique cover of G. α(G):the number of vertices in an independent set of maximum cardinality. χ(G): chromatic number of G.

8 definition(2) Observe: ω(G) ≦ χ(G), α(G) ≦ κ(G) ω(G) = α(G bar ), χ(G) = κ(G bar ) Some parameters on G1 ω(G) = 3, α(G) = 2, χ(G) =3, κ(G) = 2 Deciding ω(G), χ(G), α(G) andκ(G) are NP-hard problems. G1

9 definition(3) Separator: given a graph G=(V,E), a set of vertices S V is a separator if the subgraph of G induced by V-S is disconnected. The set S is a uv-separator if u and v are in different connected components of G[V-S]. A uv-separator S is minimal if no subset of S separates u and v.

10 Tree decomposition

11 example for example: a trivial tree decomposition contains all vertices of the graph in its single root node. The tree decomposition Is far from unique.

12 Treewidth(1) a parameter that give a measure of how “tree-like” or “close to being a tree” a graph is. The width of a tree decomposition is the size of its largest set Xi minus one. The treewidth tw(G) of a graph G is the minimum width among all possible tree decompositions of G.

13 Treewidth(2) A tree decomposition of width equal to the treewidth is called an optimal tree decomposition. For graphs that have treewidth bounded by a constant, their treewidth and corresponding optimal tree decoposition can be constructed in linear time. Since we need an optimal tree composition when designing polynomial time algorithms for graphs of bounded treewidth.

14 Dynamic programming on a tree decomposition When we have a graph of treewidth at most k, we can first compute its treewidth and an optimal tree decomposition in linear time. We turn this tree decomposition into a binary tree decomposition with the same width in polynomial time. Therefore,We can always assume that we have an optimal binary tree decomposition.

15 DP based on a tree decomposition

16 K tree

17 Partial k-tree Definition A graph G is called a partial k-tree if G is a subgraph of a k-tree. Theorem G is a partial k-tree iff G has treewidth at most k.

18 Chordal graphs(1) Definition A vertex is called simplicial if it’s adjacency set induces a clique. Theorem A graph G is chordal iff every minimal separator of G is a clique. Theorem A graph is cordal iff it has a p.e.o.

19 Chordal graphs(2) Theorem A chordal graph is either complete or has at least two nonadjacent simplicial vertices. proof Let G be a chordal graph which is not complete. By induction on vertex n. abS A B S

20 Clique tree(1) Definition There exists a tree T=( κ,ε ) whose vertex set is the set of maximal cliques of G such that each of the induced subgraphs T[ κ v ] is connected.

21 Clique tree(2)

22 Clique tree(3) Any clique tree of a chordal graph G is a tree decomposition of G of minimum width. Thus for a chordal graph G, the treewidth is one less than the size of the largest clique in G, and hence can be found in linear time. The clique tree is a useful structure to express the information on maximal cliques and minimal separators of chordal graph. Maximum clique can be found in linear time by a modification of Maximum Cardinality Search(MCS).

23 Interval graphs Definition A graph G=(V,E) is an interval graph if there is a mapping I of the vertices of G into sets of consecutive integers such that for each pair of vertices v,w ∈ E I(v) I(w) Φ Theorem G is an intergval graph if and only if G has a clique tree that is a simple path.

24 Interval graphs

25 conclusion a tree decomposition is a mapping of a graph into a tree that can be used to speed up solving certain problems on the original graph. Interval graphs have at least as many applications as chordal graphs. Many scheduling problems can be modeled as interval graphs. Actually, when G is chordal, also χ(G), α(G),κ(G) can be computed in linear time.

26 Reference: Graph Classes A Survey Lecture notes for Graph-Theoretic Algorithms, Waterloo Unv. Hung-Lin Fu, graph decomposition load Hans L. Bodlaender, A partial k-arboretum of graphs with bounded treewidth