A theorem of Ore and self-stabilizing algorithms for disjoint minimal dominating sets

Abstract

A theorem of Ore [20] states that if D is a minimal dominating set in a graph G = (V , E) having no isolated nodes, then V − D is a dominating set. It follows that such graphs must have two disjoint minimal dominating sets R and B. We describe a self-stabilizing algorithm for finding such a pair of sets. It also follows from Ore's theorem that in a graph with no isolates, one can find disjoint sets R and B where R is maximal independent and B is minimal dominating. We describe a self-stabilizing algorithm for finding such a pair. Both algorithms are described using the Distance-2 model, but can be converted to the usual Distance-1 model [7], yielding running times of O (n 2 m).

A self‐stabilizing distributed algorithm for the local (1,|Ni|)‐critical section problem

“…Additionally, it entails a burden on memory resources, higher traffic costs, and a long computational time. In the works of Hedetniemi et al 25 and Kamei and Kakugawa, 26 self‐stabilizing algorithms for finding two disjoint dominating sets are proposed, but both of their dominating sets are static.…”

“…Additionally, it entails a burden on memory resources, higher traffic costs, and a long computational time. In the works of Hedetniemi et al 25…”