Dominion of some graphs
Julian Allagan, Benkam Bobga

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.
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Taxonomy
TopicsAdvanced Graph Theory Research · Graph Labeling and Dimension Problems · Graph theory and applications
