Large-Treewidth Graph Decompositions

Treewidth is an important and a widely used graph parameter. Informally, the treewidth of a graph measures how close the graph is to being a tree. In particular, low-treewidth graphs often exhibit behavior somewhat similar to that of trees, in that many problems can be solved efficiently on such graphs, often by using dynamic programming. The treewidth of a graph G D .V;E/ is typically defined via tree decompositions. A tree decomposition for G consists of a tree T D .V .T /; E.T // and a collection of sets fXv V gv2V.T / called bags, such that the following two properties are satisfied: (i) for each edge .a; b/ 2 E, there is some node v 2 V.T / with both a; b 2 Xv , and (ii) for each vertex a 2 V , the set of all nodes of T whose bags contain a form a nonempty (connected) subtree of T . The width of a given tree decomposition is maxv2V.T /fjXvj 1g, and the treewidth of a graph G, denoted by tw.G/, is the width of a minimumwidth tree decomposition for G. In large-treewidth graph decompositions, we seek to partition a given graph G into a large number of disjoint subgraphs G1; : : : ; Gh, where each subgraph Gi has a large treewidth. Specifically, if k denotes the treewidth of G, h is the desired number of the subgraphs in the decomposition, and r is the desired lower bound on the treewidth of each subgraph Gi , then we are interested in efficient algorithms that partition any input graph G of treewidth k into h disjoint subgraphs of treewidth at least r each, and in establishing the bounds on h and r in terms of k, for which such a partition exists.