Infinite induced-saturated graphs

TL;DR

This paper proves the existence of countably infinite graphs that are $H$-induced-saturated for all finite graphs $H$ except cliques or independent sets, highlighting a strong form of induced saturation.

Contribution

It establishes the existence of countably infinite $H$-induced-saturated graphs for all non-clique, non-independent finite graphs $H$, and demonstrates a stronger property involving local modifications.

Findings

Existence of countable $H$-induced-saturated graphs for all relevant $H$

Construction of a graph where any local finite change introduces $H$

Strengthening of induced saturation concept for infinite graphs

Abstract

A graph $G$ is $H$-induced-saturated if $G$ is $H$-free but deleting any edge or adding any edge creates an induced copy of $H$. There are non-trivial graphs $H$, such as $P_4$, for which no finite $H$-induced-saturated graph $G$ exists. We show that for every finite graph $H$ that is not a clique or an independent set, there always exists a countable $H$-induced-saturated graph. In fact, we show that a far stronger property can be achieved: there is a countably infinite $H$-free graph $G$ such that any graph $G'\ne G$ obtained by making a locally finite set of changes to $G$ contains a copy of $H$.