Independent Roman Domination: The Complexity and Linear-Time Algorithm for Trees

Abstract

For a graph G=(V,E), an independent Roman dominating function (IRDF) is a function f:V→{0,1,2} having the property that: (1) every vertex assigned a value of 0 is adjacent to at least one vertex assigned a value of 2, (2) there are no two adjacent vertices with positive assignments. The weight of an IRDF (w(f)) is the sum of assignments for all vertices. The minimum weight of an independent Roman dominating function on graph G is the independent Roman domination number, denoted by iR(G). In this paper, we prove that the decision problem of minimum IRDF is NP-complete for chordal bipartite graphs. Then, we research the difference in complexity between the decision problem of RDF and IRDF. Finally, we propose a linear-time algorithm for computing the minimum weight of an independent Roman dominating function in trees.