Dominion of some graphs

TL;DR

This paper investigates the concept of dominion in graphs, specifically focusing on the number of minimum dominating sets and establishing relations for paths, cycles, and graph joins.

Contribution

It introduces the dominion parameter of graphs and derives formulas for paths, cycles, and joins, expanding understanding of dominating set structures.

Findings

Calculated dominions for paths and cycles.

Derived relations between dominion and minimum dominating sets.

Analyzed dominion in graph joins.

Abstract

Given a graph G equals (V,E), a subset S subset of V is a dominating set if every vertex in V minus S is adjacent to some vertex in S. The dominating set with the least cardinality, gamma, is called a gamma-set which is commonly known as a minimum dominating set. The dominion of a graph G, denoted by zeta(G), is the number of its gamma-sets. Some relations between these two seemingly distinct parameters are established. In particular, we present the dominions of paths, some cycles and the join of any two graphs.