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Maximal double Roman domination in graphs
Applied Mathematics and Computation  (IF4.091),  Pub Date : 2021-10-01, DOI: 10.1016/j.amc.2021.126662
H. Abdollahzadeh Ahangar, M. Chellali, S.M. Sheikholeslami, J.C. Valenzuela-Tripodoro

A maximal double Roman dominating function (MDRDF) on a graph G=(V,E) is a function f:V(G){0,1,2,3} such that (i) every vertex v with f(v)=0 is adjacent to least two vertices assigned 2 or to at least one vertex assigned 3, (ii) every vertex v with f(v)=1 is adjacent to at least one vertex assigned 2 or 3 and (iii) the set {wV|f(w)=0} is not a dominating set of G. The weight of a MDRDF is the sum of its function values over all vertices, and the maximal double Roman domination number γdRm(G) is the minimum weight of an MDRDF on G. In this paper, we initiate the study of maximal double Roman domination. We first show that the problem of determining γdRm(G) is NP-complete for bipartite, chordal and planar graphs. But it is solvable in linear time for bounded clique-width graphs including trees, cographs and distance-hereditary graphs. Moreover, we establish various relationships relating γdRm(G) to some domination parameters. For the class of trees, we show that for every tree T of order n4, γdRm(T)54n and we characterize all trees attaining the bound. Finally, the exact values of γdRm(G) are given for paths and cycles.