Extended Chain of Domination Parameters in Graphs

A subset of the vertex set of a graph is called an isolate set if the subgraph induced by has an isolated vertex. The subset is called an isolate dominating set if it is both isolate and dominating. Also, is called an isolate irredundant set if it is both isolate and irredundant. In this paper, we establish a chain connecting various isolate parameters with the existing domination parameters and discuss equality among the parameters in the extended chain. 1. Introduction One of the fastest growing areas within graph theory is the study of domination and related subset problems such as independence, covering, and matching. In fact, there are scores of graph theoretic concepts involving domination, covering, and independence. The bibliography in domination maintained by Haynes et al. [1] has over 1200 entries in which one can find an appendix listing 75 different types of domination and domination related parameters which have been studied in the literature. Hedetneimi and Laskar [2] edited the recent issue of discrete mathematics devoted entirely to domination, and a survey of advanced topics in domination is given in the book by Haynes et al. [3]. In 1978, Cockayne et al. [4] first defined what has now become a well-known inequality chain of domination related parameters of a graph as follows: where and denote the lower and upper irredundance numbers, and denote the lower and upper domination numbers, and and denote the independent domination number and independence number of a graph , respectively. Since then, more than 100 research papers have been published in which this inequality chain is the focus of study. More specifically, extending this chain in either side by some fundamental domination parameters is one such direction of research. By fundamental domination parameter, we mean that they can be defined for all nontrivial connected graphs. Following this we introduced such a new variation of domination in the name of isolate domination of graphs. By isolate dominating set of , we mean a dominating set of such that , that is , has an isolated vertex, and the corresponding parameters and are, respectively, the minimum and maximum cardinalities of a minimal isolate dominating set of . The study of this new variation of parameter was initiated in [5], where the existing domination chain (1) was extended as follows: This paper extends the above chain with the addition of some new variates related to isolate domination and discusses the relationship among some of the parameters involved in this new chain. 2. Definitions, Notations, and Preliminary