Advanced Graph Theoretical Topics Delp ensum I238 - Institutt for informatikk Pinar Heggernes August 23, 2001 Many graph problems that are NP-complete on general graphs, have p oly- nomial time solutions for sp ecial graph classes. In this do cument, a number of such graph classes are reviewed. In addition, some imp ortant graph theoretical de nitions and notions are mentioned and explained. 1 Subsets, decomp ositions, and intersections De nition 1.1 Given a graph G =(V; E) and a subset U V , the subgraph of G induced by U is the graph G[U ]= (U; D ), where (u; v ) 2 D if and only if u; v 2 U and (u; v ) 2 E . De nition 1.2 A tree-decomp osition of a graph G =(V; E) is a pair (fX j i 2 I g ; T =(I; M )) i where fX j i 2 I g isacol lection of subsets of V , and T isatree, such that: i S X = V i i2I (u; v ) 2 E )9i 2 I with u; v 2 X i For al l vertices v 2 V , fi 2 I j v 2 X g induces a connected subtreeof T . i The last condition of De nition 1.2 can b e replaced by the following equiv- alent condition: i; k ; j 2 I and j is on the path from i to k in T ) X \ X X . i k j Lemma 1.3 [6] Let (fX j i 2 I g ; T =(I; M )) be a tree decomposition of i G =(V; E), and let K V be a clique in G. Then there exists an i 2 I with K X . i The width of a decomp osition (fX j i 2 I g ; T =(I; M )) is max jX j1. i i2I i The treewidth of a graph G is the minimum width over all tree-decomp ositions of G. 1 Corollary 1.4 The treewidth of a graph G is at least one less than the size of the largest clique in G. A path-decomposition is a tree-decomp osition (fX j i 2 I g ; T =(I; M )) i such that T is a path. The pathwidth of a graph G is the minimum width over all path-decomp ositions of G. Example 1.5 Let G be the graph shown in Figure 1 a). Let I = f1; 2; 3; 4g;X = fa; b; f g;X = fb; d; f g;X = fb; c; dg;X = 1 2 3 4 fd; e; f g. Let T be the tree shown in Figure 1 b). (fX j i 2 I g;T) is a tree- i decomposition of G. Let J = f1; 2; 3g;Y = fa; b; f g;Y = fb; c; e; f g;Y = fc; d; eg. Let P be the 1 2 3 path shown in Figure1c). (fY j j 2 J g;P) is a path-decomposition of G. j a 1 1 b f 2 2 c e 3 4 d 3 a) b) c) Figure 1: The graphs of Example 1.5. The graph G of Example 1.5 has treewidth 2 and pathwidth 2 (why?). Since every path decomp osition is also a tree decomp osition, pathwidth treewidth for all graphs. Theorem 1.6 [1] The fol lowing problems are NP-complete: Given a graph G = (V; E) and an integer c < jV j, is the treewidth of G c? Given a graph G = (V; E) and an integer c < jV j, is the pathwidth of G c? The c-treewidth problem can b e solved in p olynomial time [1]: Given a graph G, is the treewidth of G at most c? In this case, c is a constant and not a part of the input. We end this section with the de nition of a minimal separator, which is central in coming sections. 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 di erent connected comp onents of G[V S ]. A uv -separator S is minimal if no subset of S separates u and v . 2 De nition 1.7 S is a minimal separator of G if there exist two vertices u and v in G such that S is a minimal uv -separator. 2 Partial k -trees De nition 2.1 The class of k -trees is de nedrecursively as fol lows: The complete graph on k vertices i a k -tree. A k -tree G with n +1 vertices (n k )can beconstructedfrom a k -tree H with n vertices by adding a vertex adjacent to exactly k vertices, namely al l vertices of a k -clique of H . The following theorem gives several alternativecharacterizations of k -trees. Theorem 2.2 [15] Let G = (V; E) be a graph. The fol lowing statements are equivalent: G is a k -tree. G is connected, G has a k -clique, but no (k +2)-cliques, and every minimal separator of G is a k -clique. 1 k (k +1), and every minimal separator of G is connected, jE j = k jV j 2 G is a k -clique. G has a k -clique, but not a (k +2)-clique, and every minimal separator of G is a clique, and for al l distinct non-adjacent pairs of vertices x; y 2 V , there exist exactly k vertex disjoint paths from x to y . De nition 2.3 A partial k -tree is a graph that contains al l the vertices and a subset of the edges of a k -tree. Theorem 2.4 [17] G isapartial k -tree if and only if G has treewidth at most k . Pro of. ): Let G b e a partial k -tree. We can assume that G is a k -tree since the treewidth cannot increase for subgraphs. Let thus G =(V; E) be a k -tree with jV j >k+1. There is avertex v 2 V such that G[V fv g] is a k -tree, and the neighb ors of v induce a clique K of size k in G. Using induction, we can assume that G[V fv g] has treewidth at most k with a corresp onding tree decomp osition (fX j i 2 I g;T =(I; M )). (The base case of induction is when i G[V fv g] is a complete graph on k vertices; such a graph has clearly treewidth 0 0 containing all neighb ors of v k 1.) By Lemma 1.3, there is an i 2 I with X i 0 in G, with K X . Let J = I [fj g, where j 62 I , and let X = K [fv g. Now, i j 0 (fX j i 2 J g;T =(J;M [fi ;jg)) is a tree decomp osition of G with width k . i (: Let G =(V; E) b e a graph with jV j >k+1,andlet(fX j i 2 I g;T = i (I; M )) b e a tree decomp osition of G of width at most k . Wewillprove, with 3 0 induction on jV j, that there is a k -tree H =(V; E )such that for every i 2 I , G[X ] is a subgraph of a clique of H with k +1 vertices. If jV j = k + 1 then we i are done. Otherwise, take a leaf no de l 2 I and let j 2 I b e the only neighbor of l 2 T . If X X , then we can remove l from T and continue with the l j remaining tree decomp osition. Let v b e the only vertex in X X (otherwise l j X can b e divided into several tree no des such that the resulting new leaf no de l satis es this requirement). Note that v do es not app ear in any X for i 6= l . i Thus all neighb ors of v must app ear in X . Remember that jX jk +1,thus v l l has at most k neighb ors. Supp ose that the induction hyp othesis on G[V fv g] 0 with tree decomp osition (fX gji 2 I; i 6= l g;T[I fl g]) results in a k -tree H . i Since v 62 X , all neighb ors of v must b elong to X . Therefore, the neighb ors of j j 0 v induce a subgraph of a k -clique C of H in G. Now, we can add v with edges 0 to all vertices of C to H (whichwillmakea (k + 1)-clique), and get the desired k -tree H . 2 Several problems that are NP-complete on general graphs have p olynomial time algorithms for partial k -trees. We will lo ok at one such example, INDE- PENDENT SET [5]. In this problem we are lo oking for the maximum size of a set X V in a graph G =(V; E)such thatnotwovertices b elonging to X are neighbors in G. Given a tree decomp osition of G, it is easy to make one that is binary with the same treewidth. Supp ose that we have a binary tree decomp osition (fX j i 2 I g;T =(I; M )) of input graph G, with ro ot r and treewidth k . For i each i 2 I , let Y = fv 2 X j j = i or j is a descendantof ig.