Abstract
Dominating sets find application in a variety of networks. A subset of nodes D is a (1,2)-dominating set in a graph G=(V,E) if every node not in D is adjacent to a node in D and is also at most a distance of 2 to another node from D. In networks, (1,2)-dominating sets have a higher fault tolerance and provide a higher reliability of services in case of failure. However, finding such the smallest set is NP-hard. In this paper, we propose a polynomial time algorithm finding a minimal (1,2)-dominating set, Minimal_12_Set. We test the proposed algorithm in network models such as trees, geometric random graphs, random graphs and cubic graphs, and we show that the sets of nodes returned by the Minimal_12_Set are in general smaller than sets consisting of nodes chosen randomly.
Highlights
IntroductionAlgorithm for Minimal (1,2)-Dominating Set in Networks
Since (1,2)-dominating sets have potential for many applications in real-life situations and up to now there is a little known about algorithms finding these sets, in our paper we propose a polynomial time algorithm that finds a minimal (1,2)-dominating set in any graph
Since determining the exact value of (1,2)-domination number is known to be NPhard for chordal bipartite graphs, C4 -free graphs, maximum degree 4 graphs, partial grid graphs and planar graphs, in this paper we propose an algorithm that finds a minimal (1,2)-dominating set in an arbitrary graph in a polynomial time
Summary
Algorithm for Minimal (1,2)-Dominating Set in Networks. Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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