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Are unique subgraphs not easier to find? (English) Zbl 06855757
Summary: Consider a pattern graph $$H$$ with $$l$$ edges, and a host graph $$G$$ which may contain several occurrences of $$H$$. In [15], we claimed that the time complexity of the problems of finding an occurrence of $$H$$ (if any) in $$G$$ as well as that of the decision version of the problem are within a multiplicative factor $$\widetilde{O}(l^3)$$ of the time complexity for the corresponding problem, where the host graph is guaranteed to contain at most one occurrence of a subgraph isomorphic to $$H$$, and the notation $$\widetilde{O}()$$ suppresses polylogarithmic in $$n$$ factors. We show a counterexample to this too strong claim and correct it by providing an $$\widetilde{O}((l(d-1)+2)^l)$$ bound on the multiplicative factor instead, where $$d$$ is the maximum number of occurrences of $$H$$ that can share the same edge in the input host graph. We provide also an analogous correction in the induced case when occurrences of induced subgraphs isomorphic to $$H$$ are sought.
##### MSC:
 68Q Theory of computing
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##### References:
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