Polynomial-time algorithms for Subgraph Isomorphism in small graph classes of perfect graphs

Abstract

a b s t r a c tGiven two graphs, Subgraph Isomorphism is the problem of deciding whether the first graph (the base graph) contains a subgraph isomorphic to the second one (the pattern graph). This problem is NP-complete even 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 cochain 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. (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.

ISOMORPH: an efficient application on GPU for detecting graph isomorphism

“…From a theoretical point of view, we have found works such as Czajka and Pandurangan (2008), Arvind et al (2012) and He et al (2021), on which this article is based, to efficiently implement this algorithm using the HPC techniques. Other future opportunities are opened from this work, for example, the sub-graph isomorphism, in the beginning also explored from a theoretical point of view, as other authors do in Cordella et al (2004), Konagaya et al (2016), but in the same way, underexplored from an efficiency point of view.…”

“…Other future opportunities are opened from this work, for example, the sub-graph isomorphism, in the beginning also explored from a theoretical point of view, as other authors do in Cordella et al . (2004), Konagaya et al . (2016), but in the same way, underexplored from an efficiency point of view.…”

Subgraph Isomorphism on Graph Classes that Exclude a Substructure

“…When the pattern graph has bounded treewidth, the celebrated color-coding technique [1] gives a fixed-parameter algorithm parameterized by the size of the pattern graph. It is also known that for chain graphs, co-chain graphs, and threshold graphs, Subgraph Isomorphism is polynomial-time solvable [20,22,21]. In the case where only the pattern graph has to be in a restricted graph class that is closed under vertex deletions, a complexity dichotomy with respect to the graph class is known [18].…”

Subgraph Isomorphism on Graph Classes that Exclude a Substructure

“…) Since most of the well-studied graph classes contain all linear forests or all cluster graphs, it is often hopeless to have a polynomial-time algorithm for an interesting graph class. This is sometimes true even if we further assume that the graphs are connected [20,22]. On the other hand, it is polynomial-time solvable for trees [29].…”

“…When the pattern graph has constant treewidth, the celebrated color-coding technique [1] gives a fixed-parameter algorithm parameterized by the size of the pattern graph. It is also known that for chain graphs, co-chain graphs, and threshold graphs, Subgraph Isomorphism is polynomial-time solvable [20][21][22]. In the case where only the pattern graph has to be in a restricted graph class that is closed under vertex deletions, a complexity dichotomy with respect to the graph class is known [18].…”