On Asymmetric Induced Saturation

Abstract

Given a graph H, a graph G is H-free if no induced subgraph of G is isomorphic to H. A graph G is H-induced-saturated if G is H-free but deleting or adding any edge in G creates an induced copy of H. The notion of induced saturation originated in a 2012 work by Martin and Smith [23] concerning the extremal properties of H-induced-saturated graphs. On the structural side, a large body of work since then has been devoted to the study of graphs H for which H-induced-saturated graphs do exist in the first place. We say that H is normal if there exists an H-induced-saturated graph. It is immediate that complete graphs (except when on two vertices) are not normal because deleting edges from a graph cannot increase its clique number. Similarly, empty graphs (except when on two vertices) are not normal because adding edges to a graph cannot increase its independence number. Beyond these trivial cases, however, characterizing normal graphs is quite difficult: The four-vertex path is the only other graph currently known not to be normal, and very few graphs are known to be normal. In particular, it remains open whether all even cycles are normal. In this thesis, we study the analogous notions with only one of the two operations – edge deletion or addition – required to create an induced copy of H. Given a graph H, we say that a graph G is H-deletion-saturated if G is H-free, has at least one edge, and deleting any edge in G creates an induced copy of H. We say that H is deletion-normal if such a graph G exists. (The complementary notions of H-addition-saturated and addition-normal are defined similarly.) These “asymmetric” weakenings of induced saturation appear to be more tractable. For example, in contrast to the fact that, as mentioned above, it remains open whether all even cycles are normal, Tennenhouse [28] proved that all even cycles are addition-normal, and with Fan, Sepehr Hajebi, and Spirkl, we proved recently [16] that all even cycles are deletion-normal. We conjecture that every non-complete graph H is deletion-normal (or equivalently, that every non-empty graph H is addition-normal), and provide evidence for our conjecture by proving it for a variety of graphs H, including all complete multipartite graphs with a unique largest part, line graphs of all trees, all triangle-free graphs with exactly one cycle, and all graphs on at most six vertices.