On Roman Domination of Graphs Using a Genetic Algorithm

Abstract

A Roman dominating function (RDF) on a graph G is a labelling f : V → {0, 1, 2} such that every vertex labelled 0 has at least one neighbour with label 2. The weight of G is the sum of the labels assigned. Roman domination number (RDN) of G, denoted by γ R (G), is the minimum of the weights of G over all possible RDFs. Finding RDN for a graph is an NP-hard problem. Approximation algorithms and bounds have been identified for general graphs and exact results exist in the literature for some standard classes of graphs such as paths, cycles, star graphs and 2 × n grids, but no algorithm has been proposed for the problem for exact results on general graphs in the literature reviewed by us. In this paper, a genetic algorithm has been proposed for the Roman domination problem in which two construction heuristics have been designed to generate the initial population, a problem specific crossover operator has been developed, and a feasibility function has been employed to maintain the feasibility of solutions obtained from the crossover operator. Experiments have been conducted on different types of graphs with known optimal results and on 120 instances of Harwell-Boeing graphs for which bounds are known. The algorithm achieves the exact RDN for paths, cycles, star graphs and 2 × n grids. For Harwell-Boeing graphs, the results obtained lie well within bounds.

The Signed (Total) Roman Domination Problem on some Classes of Planar Graphs -- Convex Polytopes

“…The basic problem in this field, the Roman domination problem (RDP), was introduced by Cockayne et al in [9]. Since then, it was atracted by many scientists to theoretically study this problem on many special classes of graphs and graph structures, see [27,36,23,22,26,30,25,21], as well as algorithmically, see [17,19,20]. RDP problem for some classes of planar graphs called convex polytopes is studied in [18].…”