Graphs with induced-saturation number zero

Sarah Behrens¹ ; Catherine Erbes² ; Michael Santana³ ; Derrek Yager³ ; Elyse Yeager⁴

¹Epic Systems Corporation
²Hiram College
³University of Illinois at Urbana-Champaign
⁴University of British Columbia

The electronic journal of combinatorics, Tome 23 (2016) no. 1

Résumé

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.

Détail

DOI : 10.37236/5095

Classification : 05C35, 05C75
Mots-clés : graph saturation, induced subgraph