Finding Induced Subgraphs via Minimal Triangulations

TL;DR

This paper introduces algorithms leveraging potential maximal cliques and minimal separators to efficiently find maximum induced subgraphs of bounded treewidth and to detect specific subgraphs, improving computational complexity.

Contribution

It applies combinatorial objects from minimal triangulations to develop fixed-parameter algorithms for subgraph detection and treewidth problems, with improved runtime bounds.

Findings

Algorithms run in time |Pi_G| * n^{O(t)} for maximum induced subgraph of treewidth t

Detection of subgraphs isomorphic to a given graph F is feasible within similar bounds

Overall runtime improved to 1.734601^n * n^{O(t)} for these problems

Abstract

Potential maximal cliques and minimal separators are combinatorial objects which were introduced and studied in the realm of minimal triangulations problems including 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 Pi_G, and an integer t, it is possible in time |Pi_G| * n^(O(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(1.734601^n), this yields that both problems are solvable in time 1.734601^n * n^(O(t)).