Some New Results on Global Dominating Sets

Abstract

A dominating set is called a global dominating set if it is a dominating set of a graph G and its complement G¯. A natural question arises: are there any graphs for which it is possible to relate the domination number and the global domination number? We have found an affirmative answer to this question and obtained some graphs having such characteristic.

Eccentric domination number of some path related graphs

“…There are various domination models available in the literature. Total domination [3], equitable domination [7], global domination [10], steiner domination [9], independent domination [4], restrained domination [8], eccentric domination [6] are among worth to mention.…”

Restrained Domination Number of Some Path Related Graphs

“…A set S ∈ V (G) is a restrained dominating set if every vertex in V (G) -S is adjacent to a vertex in S and to a vertex in V (G) -S. The restrained domination number of G, denoted by γ r (G) , is the minimum cardinality of a restrained dominating set of G. There are various domination models available in the literature. Some of them are total domination [3], equitable domination [11], global domination [13], steiner domination [14], independent domination [1] ISSN: 2456-8686, Volume 1, Issue 1, 2017:123-128 DOI : http://doi.org/10.26524/cm10 are among worth to mention. The concept of restrained domination was conceived by Telle and Proskurowski [12] as a vertex partitioning problem.…”

Some results on global dominating sets

“…The various bounds of global domination number in terms of order, degrees and domination number of graph is given by Brigham and Duttom [1]. The global domination number for one-point union of finite number of copies of cycles C n is reported in Vaidya and Pandit [5] while Kulli and Janakiram [3] have introduced the concept of total global dominating sets.…”