The Vertex-Edge Locating Roman Domination of Some Graphs

In this paper, we introduce the concept of vertex-edge locating Roman dominating functions in graphs. A vertex-edge locating Roman dominating (\({ve} - {LRD}\)) function of a graph \(G = {(V,E)}\) is a function \(f:{{V{(G)}}\rightarrow{\{ 0,1,2\}}}\) such that the following conditions are satisfied: (i) for every adjacent vertices \(u,v\) with \({f{(u)}} = 0\) or \({f{(v)}} = 0\), there exists a vertex \(w\) at distance \(1\) or \(2\) from \(u\) or \(v\) with \({f{(w)}} = 2\), (ii) for every edge \({uv} \in E\), \({max{\lbrack{f{(u)}},{f{(v)}}\rbrack}} \neq 0\) and (iii) any pair of distinct vertices \(u,v\) with \({f{(u)}} = {f{(v)}} = 0\) does not have a common neighbour \(w\) with \({f{(w)}} = 2\) . The weight of \(ve\)-LRD function is the sum of its function values over all the vertices. The vertex-edge locating Roman domination number of \(G\) denoted by \(\gamma_{{ve} - {LR}}^{P}{(G)}\) is the minimum weight of a \(ve\)-LRD function in \(G\). We proved that the vertex-edge locating Roman domination problem is NP complete for bipartite graphs. Also, we present the upper and lower bonds of \(ve\)-LRD function for trees. Lastly, we give the upper bounds of \(ve\)-LRD function for some connected graphs.