Induced saturation of graphs

Abstract

A graph G is H-saturated for a graph H, if G does not contain a copy of H but adding any new edge to G results in such a copy. An H-saturated graph on a given number of vertices always exists and the properties of such graphs, for example their highest density, have been studied intensively.A graph G is H-induced-saturated if G does not have an induced subgraph isomorphic to H, but adding an edge to G from its complement or deleting an edge from G results in an induced copy of H. It is not immediate anymore that H-induced-saturated graphs exist. In fact, Martin and Smith (2012) showed that there is no P 4 -induced-saturated graph. Behrens et al. (2016) proved that if H belongs to a few simple classes of graphs such as a class of odd cycles of length at least 5, stars of size at least 2, or matchings of size at least 2, then there is an H-induced-saturated graph.This paper addresses the existence question for H-induced-saturated graphs. It is shown that Cartesian products of cliques are H-induced-saturated graphs for H in several infinite families, including large families of trees. A complete characterization of all connected graphs H for which a Cartesian product of two cliques is an H-induced-saturated graph is given. Finally, several results on induced saturation for prime graphs and families of graphs are provided.

Induced saturation of P6

“…Unfortunately the construction gives no idea on what happens for other values of n. Note that the conditions 'removing any edge from G gives an induced P n ' and 'adding any missing edge to G gives an induced P n ' can be together written as 'there exists f : V (G) (2) → V (G) (n−2) such that for any e ∈ V (G) (2) , we have e∩f (e) = ∅ and G e [e ∪ f (e)] is isomorphic to P n ', where G e is the graph obtained from G by removing or adding e depending on whether e ∈ E (G) or e ∈ E (G) respectively. If m = |V (G)|, then f can be viewed as a function f : [m] (2) → [m] (n−2) . Even though the condition that G does not contain induced copy of P n is not yet considered, this suggests that the other conditions should be easier to satisfy when n increases.…”

“…It is natural to ask what happens when H = P n for other values of n. The cases H = P 2 and H = P 3 are trivial, as one can take G to be empty graph and K m respectively. Axenovich and Csikós [2] investigated several related questions by giving examples of families of graphs that are induced-saturated, and also asked if the graphs H = P n are induced-saturated for n ≥ 5. The aim of this note is to provide an example which shows that P 6 is inducedsaturated.…”

$P_{n}$-Induced-Saturated Graphs Exist for all $n \geqslant 6$

“…In 2012, Martin and Smith [6] introduced the notion of induced saturation on trigraphs. As a special case of this more general framework, there arises the notion of inducedsaturated graphs, first studied in its own right by Behrens et al [2] and later also by Axenovich and Csikós [1]. Given graphs G, H, we say G is H-induced-saturated if G contains no induced subgraph isomorphic to H, but deleting any edge of G creates an induced subgraph isomorphic to H, and adding any new edge to G from G c also creates an induced subgraph isomorphic to H. Throughout the rest of the note, we will abbreviate a H-induced-saturated graph as a H-IS graph.…”

“…On the other hand, it is easy to see that there do exist P 2 -IS and P 3 -IS graphs. This leads to a question, asked by Axenovich and Csikós [1], for what integers n 5 do there exist P n -IS graphs. Räty [7] was the first to make a progress on this question, showing by an algebraic construction that there exists a P 6 -IS graph.…”

On induced saturation for paths

“…Recently, in 2019, Axenovich and Csikós [1] introduced the notion of an H-induced-saturated graph, whose definition emphasizes that a non-edge is as important as an edge when considering induced subgraphs. For a graph H, a graph G is H-induced-saturated if G does not contain an induced copy of H, but either removing an edge from G or adding a non-edge to G creates an induced copy of H. In contrast to the fact that an H-saturated graph always exists for an arbitrary graph H, it is not always the case that an H-induced-saturated graph exists.…”