Independent Roman bondage of graphs

An independent Roman dominating function (IRD-function) on a graph G is f : V ( ) → {0, 1, 2} satisfying the conditions that (i) every vertex u for which = 0 adjacent to at least one v 2, and (ii) set of all vertices assigned non-zero values under independent. The weight an IRD-function sum its over vertices, domination number i R minimum . In this paper, we initiate study bondage b iR having component order three, defined as smallest size edges F ⊆ E − > ). We begin by showing decision problem associated with NP-hard bipartite graphs. Then various upper bounds are established well exact it some special particular, trees T shown ≤ 3, while connected planar graphs in terms maximum degree refinements depending girth graph.