Elsevier

Discrete Applied Mathematics

Volume 285, 15 October 2020, Pages 182-187
Discrete Applied Mathematics

Note
A short note on the existence of infinite sequences of γ-graphs of graphs

https://doi.org/10.1016/j.dam.2020.05.005Get rights and content

Abstract

For a graph G=(V,E), the γ-graph of G, G(γ)=(V(γ),E(γ)), is the graph whose vertex set is the collection of minimum dominating sets, or γ-sets of G, and two γ-sets are adjacent in G(γ) if they differ by a single vertex and the two different vertices are adjacent in G. We consider sequences of γ-graphs, formed by repeatedly taking the γ-graph of a graph. By considering the γ-graphs of powers of complete graphs, we demonstrate an example claimed to be an infinite sequence of γ-graphs is incorrect, and hence open the question of the existence of such a sequence.

Introduction

Let G be a graph and let v be a vertex of G. The open neighbourhood of a vertex v, denoted N(v), is the set of vertices adjacent to v, and the closed neighbourhood of a vertex v, denoted N[v], is N(v){v}. For a subset of vertices S, we say N(S)=xSN(x) and N[S]=SN(S). A subset of vertices D is a dominating set of G if N[D]=V(G), that is, every vertex not in D is adjacent to a vertex in D. The domination number of a graph G, denoted γ(G), is the minimum cardinality of a dominating set of G. A dominating set of minimum cardinality is said to be a γ-set.

For a dominating set D of G and a vertex xD, the set of private neighbours of x, denoted pn(x,D), is N[x]N[Dx], that is, the set of vertices in the closed neighbourhood of x and not in the closed neighbourhood of any other vertex in D. If xpn(x,D), then x is a D-self private neighbour and if yx and ypn(x,D), then y is an D-external private neighbour of x. If D is a γ-set, then every vertex in D has a private neighbour.

The γ-graph of a graph G, introduced by Fricke et al. [4], is a graph denoted G(γ), has its vertex set as the γ-sets of G, and two γ-sets D1 and D2 are adjacent in G(γ) if there are vertices uD1 and vD2 such that D2=D1{u}{v} and uvE(G). Starting with D1, we think of making a swap, that is, changing vertex u for v, to form D2. A slightly different model, in which uv need not be an edge in G, was introduced independently by Subramanian and Sridharan [6]; we do not consider this model here. Fricke et al. [4] studied properties of γ-graphs, and raised seven questions, three of which remain open:

  • 1.

    Which graphs are γ-graphs of trees?

  • 2.

    For which graphs G is G(γ)G?

  • 3.

    Under what conditions is G(γ) a disconnected graph?

An algorithm for finding the γ-graph of a tree and a simple characterization of trees which are γ-graphs of a tree were given by Finbow and van Bommel [3]. Edwards, MacGillivray and Nasserasr [2] provided upper bounds on the maximum degree, the diameter, and the order of the γ-graph of a tree. Results on the structure of the γ-graphs of trees, in particular its connections with Cartesian product graphs, the set of graphs which can be obtained from the Cartesian product of graphs of order at least two is also presented in [3]. Connelly, Hutson, and Hedetneimi [1] have shown every graph is the γ-graph of a some graph.

Fricke et al. [4] present the idea of repeatedly constructing the γ-graph of a graph G, i.e., constructing the γ-graph sequence GγG(γ)γ(G(γ))(γ)γ. They note that the sequences often end with K1; however, any graph which is the γ-graph of itself is a possible ending. On the other hand, they claim the following result without proof, where GH denotes the Cartesian product of graphs G and H, defined by V(GH)=V(G)×V(H),E(GH)={((u,v),(x,w)):u=x&vwE(H), or uxE(G)&v=w}.

Claim 1.1 [4]

K3P2γK3K3γK3K3K3γ.

In Section 2, we show this claim is incorrect, and hence raise the question of whether there exists an infinite γ-graph sequence, or if all γ-graph sequences are eventually periodic. Hence, we will begin by considering the γ-graphs of small powers of complete graphs. We will use the following results throughout.

Lemma 1.2

Let G be a graph and D be a γ-set of G. If some vertex x in D has two D-external private neighbours, v and w, such that N[v]N[w]N[x]={x}, then for every neighbour F of D in G(γ), xF.

Proof

Suppose F is a γ-set of G adjacent to D. Then there exist vertices x,y such that F=D{x}{y} and xyE(G) (hence yN[x]). If v and w are D-external private neighbours of x, they are adjacent to no vertex in F{y}, so yN[v] and yN[w]. But if N[v]N[w]N[x]={x}, then y=x, which is a contradiction, and the result follows.  

Corollary 1.3

If G is a graph, D is a γ-set of G, and every vertex x in D has two D-external private neighbours, v and w, such that N[v]N[w]N[x]={x}, then D is an isolated vertex in G(γ).

Proof

Suppose F is a γ-set of G adjacent to D. For each vertex xD, it follows from Lemma 1.2 that xF. Hence, F=D, and the result follows.  

Section snippets

γ-Graphs of powers of complete graphs

Let Kn2=KnKn and for d3, let Knd=KnKnd1. In this section, we consider the γ-graphs of Kn2, Kn3, and, as a special case, K34, each of which can be used to establish Claim 1.1 is incorrect.

We begin by determining the γ-graphs of Kn2. Let KnnKnn be the graph constructed as follows. Take two copies of Knn. Consider the vertices of each copy of Knn as ordered n-tuples with two vertices adjacent if their n-tuples differ in exactly one position. Now, for every permutation π of [n]={1,2,,n}, we

CRediT authorship contribution statement

Stephen Finbow: Conceptualization, Formal analysis, Writing - review & editing, Visualization, Supervision, Funding acquisition. Rory MacIssac: Formal analysis. Christopher M. van Bommel: Conceptualization, Formal analysis, Software, Writing - original draft.

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This research was funded by Natural Sciences and Engineering Research Council of Canada Grant Number 2014-06571.

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