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

TL;DR

This paper investigates how the $c$-closure property of graphs influences the complexity of detecting and enumerating small induced subgraphs, providing parameterized algorithms for these problems.

Contribution

It systematically analyzes the complexity of subgraph detection and enumeration in $c$-closed graphs, introducing algorithms for small subgraphs based on $c$-closure.

Findings

Polynomial-time algorithms for detecting small subgraphs in $c$-closed graphs.

Complexity results vary depending on the subgraph and $c$-closure parameter.

Enhanced understanding of subgraph enumeration in structured graph classes.

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.