Halfway to induced saturation for even cycles

TL;DR

This paper advances the understanding of induced saturation by constructing graphs that are free of even cycles but become non-free upon edge removal, specifically for even cycles up to length 10.

Contribution

It provides the first constructions of $H$-induced-saturated graphs for all even cycles up to length 10, addressing a significant open problem in graph theory.

Findings

Constructed $H$-induced-saturated graphs for even cycles up to length 10

Showed existence of graphs that are $H$-free but become non-$H$-free after edge removal

Extended understanding of induced saturation for even cycles

Abstract

For graphs $G$ and $H$, we say that $G$ is $H$-free if no induced subgraph of $G$ is isomorphic to $H$, and that $G$ is $H$-induced-saturated if $G$ is $H$-free but removing or adding any edge in $G$ creates an induced copy of $H$. A full characterization of graphs $H$ for which $H$-induced-saturated graphs exist remains elusive. Even the case where $H$ is a path -- now settled by the collective results of Martin and Smith, Bonamy et al., and Dvo\'{r}\v{a}k -- was already quite challenging. What if $H$ is a cycle? The complete answer for odd cycles was given by Behren et al., leaving the case of even cycles (except for the $4$-cycle) wide open. Our main result is the first step toward closing this gap: We prove that for every even cycle $H$, there is a graph $G$ with at least one edge such that $G$ is $H$-free but removing any edge from $G$ creates an induced copy of $H$ (in fact, we construct $H$-induced-saturated graphs for every even cycle $H$ on at most 10 vertices).