Polynomial-Time Algorithms for Subgraph Isomorphism in Small Graph Classes of Perfect Graphs

Abstract

AbstractGiven two graphs, Subgraph Isomorphism is the problem of deciding whether the first graph (the base graph) contains a subgraph isomorphic to the second graph (the pattern graph). This problem is NP-complete for very restricted graph classes such as connected proper interval graphs. Only a few cases are known to be polynomial-time solvable even if we restrict the graphs to be perfect. For example, if both graphs are co-chain graphs, then the problem can be solved in linear time.In this paper, we present a polynomial-time algorithm for the case where the base graphs are chordal graphs and the pattern graphs are co-chain graphs. We also present a linear-time algorithm for the case where the base graphs are trivially perfect graphs and the pattern graphs are threshold graphs. These results answer some of the open questions of Kijima et al. [Discrete Math. 312, pp. 3164–3173, 2012]. To present a complexity contrast, we then show that even if the base graphs are somewhat restricted perfect graphs, the problem of finding a pattern graph that is a chain graph, a co-chain graph, or a threshold graph is NP-complete.KeywordsSubgraph isomorphismGraph classPolynomial-time algorithmNP-completeness