Graphs with Induced-Saturation Number Zero

Given graphs G and H, G is H-saturated if H is not a subgraph of G, but for all e / ∈ E(G), H appears as a subgraph of G + e. While for every n > |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 > |V (H)|, for which every n-vertex graph G either contains H as an induced subgraph, or there exists e / ∈ E(G) such that G + e does not contain H as an induced subgraph. To circumvent this Martin and Smith [12] 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 ∗Supported in part by National Science Foundation grant DMS-0914815. †Supported in part by National Science Foundation Grant DGE-0742434, UCD GK12 Transforming Experiences Project. ‡The research of the third, fourth, and fifth authors is supported in part by National Science Foundation grant DMS 08-38434 “EMSW21-MCTP: Research Experience for Graduate Students.” the electronic journal of combinatorics 23(1) (2016), #P1.54 1 provide bounds on indsat∗(n,H) for many graphs. In particular, we determine indsat∗(n,H) completely when H is the paw graph K1,3 + e, and we determine indsat(n,K1,3) within an additive constant of four. 1 Background and Introduction 1.1 Background and Definitions A well-known graph parameter is the saturation number, defined for a graph H and a whole number n as the minimum number of edges in a graph G on n vertices such that H is not a subgraph of G, but H occurs if any edge is added to G. Formally, sat(n,H) = min{|E(G)| : G has n vertices, H 6⊆ G, and ∀e / ∈ E(G), H ⊆ G+ e}. Determining the saturation number for a given graph H has proven, in general, quite difficult. For more information on the saturation number, see the dynamic survey of Faudree, Faudree, and Schmitt [6]. A natural attempt at defining an induced variant of graph saturation would be to state that an n-vertex graph G is H-induced-saturated if G is H-free (that is, without H as an induced subgraph) and for all e / ∈ E(G), G+ e contains H as an induced subgraph. Unfortunately, this is not always well-defined. That is, there exist graphs H and values of n > |V (H)| for which every n-vertex graph G either contains H as induced subgraph, or there exists e / ∈ E(G) such that G+e is H-free. A simple example is n = 4 and H = K1,3. In this paper, we consider a variant of the saturation number introduced by Martin and Smith in 2012 that looks for induced copies of H, and considers deleting as well as adding edges. To create a well-defined parameter, Martin and Smith [12] make use of trigraphs, objects also used by Chudnovsky and Seymour in their structure theorems on claw-free graphs [3]. Definition 1.1. A trigraph T is a quadruple (V (T ), EB(T ), EW (T ), EG(T )), where V (T ) is the vertex set and the other three elements partition ( V (T ) 2 ) into a set EB(T ) of black edges, a set EW (T ) of white edges, and a set EG(T ) of gray edges. These can be thought of as edges, nonedges, and potential edges, respectively. For any e ∈ EB(T ) ∪ EW (T ), let Te denote the trigraph where e is changed to a gray edge, i.e. Te = (V (T ), EB(T ) − e, EW (T )− e, EG(T ) + e). A realization of T is a graph G = (V (G), E(G)) with V (G) = V (T ) and E(G) = EB(T ) ∪ S for some S ⊆ EG(T ). Let R(T ) be the family of graphs that are a realization of T . A trigraph T is H-induced-saturated if no realization of T contains H as an induced subgraph, but H occurs as an induced subgraph of some realization whenever any black or white edge of T is changed to gray. The induced saturation number of a graph H with respect to n, written indsat(n,H), is the minimum number of gray edges in an H-induced-saturated trigraph with n vertices. the electronic journal of combinatorics 23(1) (2016), #P1.54 2