Computing and Listing Avoidable Vertices and Paths
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A simplicial vertex of a graph is a vertex whose neighborhood is a clique. It is known that listing all simplicial vertices can be done in O(nm) time or O(n^ω) time, where O(n^ω) is the time needed to perform a fast matrix multiplication. The notion of avoidable vertices generalizes the concept of simplicial vertices in the following way: a vertex u is avoidable if every induced path on three vertices with middle vertex u is contained in an induced cycle. We present algorithms for listing all avoidable vertices of a graph through the notion of minimal triangulations and common neighborhood detection. In particular we give algorithms with running times O(n^2m) and O(n^1+ω), respectively. However, we propose a faster algorithm that runs in time O(n^2 + m^2), and thus matches the corresponding running time of listing the simplicial vertices on sparse graphs with m=O(n). To complement our results, we consider their natural generalizations of avoidable edges and avoidable paths. We propose an O(nm)-time algorithm that recognizes whether a given induced path is avoidable.
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