Counting Graph Isomorphisms among Chordal Graphs with Restricted Clique Number

In this paper, we study the following problem: given two graphs G and H and an isomorphism ϕ between an induced subgraph of G and an induced subgraph of H, compute the number of isomorphisms between G and H that doesn't contradict to ϕ. We show that this problem can be solved in O((k + 1)!n3) time when input graphs are restricted to chordal graphs with clique number at most k + 1. To show this, we first show that the tree model of a chordal graph can be uniquely constructed except for the ordering of children of each node in O(n3) time. Then, we show that the number of isomorphisms between G and H that doesn't contradict to ϕ can be efficiently computed by use of the tree model.

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