Enumerating Simple Paths from Connected Induced Subgraphs
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We present an exact formula for enumerating the simple paths between any two vertices of a graph. Our formula involves the adjacency matrices of the connected induced subgraphs and remains valid on weighted and directed graphs. As a particular case, we obtain a relation linking the Hamiltonian paths and cycles of a graph to its dominating connected sets.
KeywordsDirected graph Self-avoiding walks Simple cycles Hamiltonian paths Dominating sets Inclusion–exclusion
P.-L. Giscard is grateful for the financial support from the Royal Commission for the Exhibition of 1851. The authors are grateful to an anonymous referee for its many constructive remarks that helped improve the paper.
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