Complexity

Volume 2018, Article ID 3041426, 8 pages

https://doi.org/10.1155/2018/3041426

## The Domination Complexity and Related Extremal Values of Large 3D Torus

^{1}Institute of Computing Science and Technology, Guangzhou University, Guangzhou 510006, China^{2}Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran^{3}Department of Mathematics, Savannah State University, Savannah, GA 31404, USA

Correspondence should be addressed to Shaohui Wang; moc.oohay@gnawiuhoahs

Received 18 January 2018; Accepted 15 April 2018; Published 2 July 2018

Academic Editor: Christos Volos

Copyright © 2018 Zehui Shao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Domination is a structural complexity of chemical molecular graphs. A dominating set in a (molecular) graph is a subset such that each vertex in is adjacent to at least one vertex in . The domination number of a graph is the minimum size of a dominating set in . In this paper, computer-aided approaches for obtaining bounds for domination number on torus graphs are here considered, and many new exact values and bounds are obtained.

#### 1. Introduction

The structure of molecular graphs is a broad area among chemistry, biology, and mathematics. In particular, domination number or dominating complexity has attracted considerable attention in the general case [1–7], like coding theory and combinatorial design. Due to a variety of applications, it has been studied on various types of graphs such as generalized Petersen graphs [8–11], hypercubes [12, 13], Fibonacci cubes [14], Kneser graphs [15–18], torus graphs [19–21], and grid graphs [22, 23]. Others are referred to [24–26].

In this work, we consider certain graphs without loops or multiple edges. A dominating set in a graph is a subset such that each vertex in is adjacent to at least one vertex in . The domination number of a graph is the minimum cardinality of a dominating set in . A dominating set of a graph is a set if it has cardinality . A dominating set is an independent dominating set, if no two vertices in are adjacent. The independent domination number of a graph is the minimum cardinality of an independent dominating set in . For , the cardinality of a minimum dominating set of containing is denoted by , that is,

For two graphs and , the *Cartesian product * is a graph with vertex set and two vertices (*,*) and (*,*) are adjacent if and only if either and or and . The gird is , the cylinder for is , the torus for and is , and the 3D torus for given , , is . The subgraph of induced by is isomorphic to . It is called a *G*-fiber and is denoted by .

The domination number has attracted considerable attention in the general case [1, 2]. Due to a variety of applications, it has been studied on various types of graphs such as generalized Petersen graphs [8–11], hypercubes [12, 13], Fibonacci cubes [14], Kneser graphs [15, 16, 18], torus graphs [19–21], and grid graphs [22, 23]. Others are referred to [24–26].

Dominating sets in some special graphs have applied in coding theory and combinatorial design. For instance, there is a natural relation between dominating sets of torus graphs and covering codes in the Lee metric [21], and dominating sets in a type of Kneser graphs are also related to Steiner system [18].

The following result is a special case of Theorem 3.3 in [19]:

Theorem 1.

Let be the automorphism group of graph . We say two subsets and are equivalent if there exists a mapping such that . For a subset , the orbit of is the family .

#### 2. The Approaches

##### 2.1. ILP-Based Search

Let be a graph and be a dominating set of . We use a Boolean variable to denote if a vertex is in , that is, if .

The following is the well-known integer linear programming model for finding a minimum dominating set of :

ILPDominatingSet

##### 2.2. Extending Dominating Sets from Inequivalent Seeds

Sometimes, we can succeed to obtain the independent domination number of a graph in a reasonable time but it fails to exhaustively search the domination number . In such cases, based on Observation 1, we can obtain first to assume there are two adjacent vertices in a minimum dominating set.

*Observation 1. *Let be a graph. If there is a minimum dominating set such that , then contains two adjacent vertices.

*Observation 2. *Let be a graph and be a family of inequivalent subsets of . If there is a minimum dominating set of containing a subset isomorphic to one element in , then .

Based on the above observations, we first select a family of inequivalent subsets of the vertex set of a considered graph then extend each subset to a minimum dominating set, that is, we compute . We apply the following procedure to compute the domination number of (using a 2.66 GHz Intel Core (TM) i7-5600U CPU).