NoteA short note on the existence of infinite sequences of -graphs of graphs☆
Introduction
Let be a graph and let be a vertex of . The open neighbourhood of a vertex , denoted , is the set of vertices adjacent to , and the closed neighbourhood of a vertex , denoted , is . For a subset of vertices , we say and . A subset of vertices is a dominating set of if , that is, every vertex not in is adjacent to a vertex in . The domination number of a graph , denoted , is the minimum cardinality of a dominating set of . A dominating set of minimum cardinality is said to be a -set.
For a dominating set of and a vertex , the set of private neighbours of , denoted , is , that is, the set of vertices in the closed neighbourhood of and not in the closed neighbourhood of any other vertex in . If , then is a -self private neighbour and if and , then is an -external private neighbour of . If is a -set, then every vertex in has a private neighbour.
The -graph of a graph , introduced by Fricke et al. [4], is a graph denoted , has its vertex set as the -sets of , and two -sets and are adjacent in if there are vertices and such that and . Starting with , we think of making a swap, that is, changing vertex for , to form . A slightly different model, in which need not be an edge in , was introduced independently by Subramanian and Sridharan [6]; we do not consider this model here. Fricke et al. [4] studied properties of -graphs, and raised seven questions, three of which remain open:
- 1.
Which graphs are -graphs of trees?
- 2.
For which graphs is ?
- 3.
Under what conditions is a disconnected graph?
An algorithm for finding the -graph of a tree and a simple characterization of trees which are -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 -graph of a tree. Results on the structure of the -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 -graph of a some graph.
Fricke et al. [4] present the idea of repeatedly constructing the -graph of a graph , i.e., constructing the -graph sequence . They note that the sequences often end with ; however, any graph which is the -graph of itself is a possible ending. On the other hand, they claim the following result without proof, where denotes the Cartesian product of graphs and , defined by
Claim 1.1 [4] .
In Section 2, we show this claim is incorrect, and hence raise the question of whether there exists an infinite -graph sequence, or if all -graph sequences are eventually periodic. Hence, we will begin by considering the -graphs of small powers of complete graphs. We will use the following results throughout.
Lemma 1.2 Let be a graph and be a -set of . If some vertex in has two -external private neighbours, and , such that , then for every neighbour of in , .
Proof Suppose is a -set of adjacent to . Then there exist vertices such that and (hence ). If and are -external private neighbours of , they are adjacent to no vertex in , so and . But if , then , which is a contradiction, and the result follows. □
Corollary 1.3 If is a graph, is a -set of , and every vertex in has two -external private neighbours, and , such that , then is an isolated vertex in .
Proof Suppose is a -set of adjacent to . For each vertex , it follows from Lemma 1.2 that . Hence, , and the result follows. □
Section snippets
-Graphs of powers of complete graphs
Let and for , let . In this section, we consider the -graphs of , , and, as a special case, , each of which can be used to establish Claim 1.1 is incorrect.
We begin by determining the -graphs of . Let be the graph constructed as follows. Take two copies of . Consider the vertices of each copy of as ordered -tuples with two vertices adjacent if their -tuples differ in exactly one position. Now, for every permutation of , 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.
References (6)
- et al.
A note on -graphs
AKCE J. Graphs Combin.
(2011) - et al.
Reconfiguring minimum dominating sets: The -graph of a tree
Discuss. Math. Graph Theory
(2018) - et al.
-Graphs of trees
Algorithms
(2019)
Cited by (0)
- ☆
This research was funded by Natural Sciences and Engineering Research Council of Canada Grant Number 2014-06571.