Detecting and Enumerating Small Induced Subgraphs in $c$-Closed Graphs

Abstract

Fox et al. [SIAM J. Comp. 2020] introduced a new parameter, called c-closure, for a parameterized study of clique enumeration problems. A graph G is c-closed if every pair of vertices with at least c common neighbors is adjacent. The c-closure of G is the smallest c such that G is c-closed. We systematically explore the impact of c-closure on the computational complexity of detecting and enumerating small induced subgraphs. More precisely, for each graph H on three or four vertices, we investigate parameterized polynomial-time algorithms for detecting H and for enumerating all occurrences of H in a given c-closed graph.