Finding Induced Subgraphs via Minimal Triangulations

Abstract

Potential maximal cliques and minimal separators are combinatorial objects which were introduced and studied in the realm of minimal triangulation problems in- cluding Minimum Fill-in and Treewidth. We discover unexpected applications of these notions to the field of moderate exponential algorithms. In particular, we show that given an n-vertex graph G together with its set of potential maximal cliques, and an integer t, it is possible in time the number of potential maximal cliques times O(nO(t)) to find a maximum induced subgraph of treewidth t in G and for a given graph F of treewidth t, to decide if G contains an induced subgraph isomorphic to F. Combined with an improved algorithm enumerating all potential maximal cliques in time O(1.734601n ), this yields that both the problems are solvable in time 1.734601n * nO(t) .