Abstract
Subgraph isomorphism is a fundamental problem in graph theory. In this paper we focus on listing subgraphs isomorphic to a given pattern graph. First, we look at the algorithm due to Chiba and Nishizeki for listing complete subgraphs of fixed size, and show that it cannot be extended to general subgraphs of fixed size. Then, we consider the algorithm due to Ga̧sieniec et al. for finding multiple witnesses of a Boolean matrix product, and use it to design a new output-sensitive algorithm for listing all triangles in a graph. As a corollary, we obtain an output-sensitive algorithm for listing subgraphs and induced subgraphs isomorphic to an arbitrary fixed pattern graph.
Highlights
The decision version of the subgraph isomorphism problem is to decide if a host graph has a subgraph isomorphic to a pattern graph
The search version asks for finding a subgraph of the host graph isomorphic to the pattern graph
The listing version asks for a list of all subgraphs of the host graph isomorphic to the pattern graph
Summary
Since there might be up to Ω(np ) subgraphs or induced subgraphs isomorphic to the pattern graph, the brute force method cannot be substantially asymptotically improved in case of the listing versions in general. This does not exclude the possibility of faster listing algorithms for restricted graph classes and output-sensitive algorithms whose complexity depends on the size of the output list of isomorphic subgraphs. We can generalize our combined algorithm to include listing of subgraphs or induced subgraphs isomorphic to an arbitrary fixed pattern graph. For a graph with n nodes and l subgraphs (induced subgraphs, respectively) isomorphic to a fixed pattern graph with e 2.3727dh/3e ) for h nodes, our combined algorithm lists all the aforementioned subgraphs in time O(n e 1.9368dh/3e l0.31754 ) for n1.3727dh/3e 6 l < n2.3940dh/3e , and O(n e 1.5dh/3e l0.5 ) for l < n1.3727dh/3e , O(n n2.3940dh/3e 6 l < n3dh/3e
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