Asymmetric induced saturation

Abstract

For which graphs $H$ does there exist a graph $G$ with at least one edge and no induced subgraph isomorphic to $H$, such that deleting any edge of $G$ creates an induced copy of $H$? We call such a graph "$H$-deletion-saturated". This version of the well-studied notion of "$H$-induced-saturated" graphs -- where both adding and deleting any edge creates an induced copy of $H$ -- appears more tractable. For example, while it remains wide open whether $H$-induced-saturated graphs exist for every even cycle $H$, we proved recently that deletion-saturated graphs exist for all even cycles. In fact, apart from complete graphs, no graph $H$ is known for which $H$-deletion-saturated graphs do not exist. We conjecture that $H$-deletion-saturated graphs exist for every non-complete graph $H$, and prove this conjecture for several types of graphs, including: complete bipartite graphs with parts of unequal size, triangle-free graphs with one cycle, graphs with two leaves at distance at most three, and line graphs of trees. In fact, in all cases, we prove the conjecture for substantially more general families. We also verify our conjecture for every graph $H$ on at most six vertices.