Graphs with Induced-Saturation Number Zero

Abstract

Given graphs $G$ and $H$, $G$ is $H$-saturated if $H$ is not a subgraph of $G$, but for all $e \notin E(G)$, $H$ appears as a subgraph of $G + e$. While for every $n \ge |V(H)|$, there exists an $n$-vertex graph that is $H$-saturated, the same does not hold for induced subgraphs. That is, there exist graphs $H$ and values of $n \ge |V(H)|$, for which every $n$-vertex graph $G$ either contains $H$ as an induced subgraph, or there exists $e \notin E(G)$ such that $G + e$ does not contain $H$ as an induced subgraph. To circumvent this Martin and Smith make use of a generalized notion of "graph" when introducing the concept of induced saturation and the induced saturation number of graphs. This allows for edges that can be included or excluded when searching for an induced copy of $H$, and the induced saturation number is the minimum number of such edges that are required.In this paper, we show that the induced saturation number of many common graphs is zero. This yields graphs that are $H$-induced-saturated. That is, graphs such that no induced copy of $H$ exists, but adding or deleting any edge creates an induced copy of $H$. We introduce a new parameter for such graphs, indsat*($n;H$), which is the minimum number of edges in an $H$-induced-saturated graph. We provide bounds on indsat*($n;H$) for many graphs. In particular, we determine indsat*($n;H$) completely when $H$ is the paw graph $K_{1,3}+e$, and we determine indsat*(n;$K_{1,3}$) within an additive constant of four.

Induced saturation of graphs

“…Using the structural descriptions of the classes Forb(C 4 , C 4 , P 4 ) (due to Chvátal and Hammer [2]) and Forb(C 4 , C 4 , C 5 ) (due to Földes and Hammer [6]), Behrens et al [1] showed that there is no graph G so that G → → {C 4 , C 4 , P 4 } or G → → {C 4 , C 4 , C 5 }. The later example shows that even though there are graphs G 1 , G 2 , G 3 so that G 1 → → C 4 , G 2 → → C 4 and G 3 → → C 5 , there is no induced-saturated graph for the family {C 4 , C 4 , C 5 }.…”

“…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.…”

“…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.…”

$P_{n}$-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 induced saturation for paths

“…Utilizing the notion of trigraphs, Prömel and Steger [12] and Martin and Smith [11] defined the induced subgraph version of the Turán number and the saturation number, respectively, of a given graph. We omit the exact definitions here, see [2,3,7,9] for more details and other recent work on Turán-type problems concerning induced subgraphs.…”