Unique identification and domination of edges in a graph: The vertex-edge dominant edge metric dimension

Abstract: Dominating sets and resolving sets have important applications in control theory and computer science. In this paper, we introduce an edge-analog of the classical dominant metric dimension of graphs. By combining the concepts of a vertex-edge dominating set and an edge resolving set, we introduce the notion of a vertex-edge dominant edge resolving set of a graph. We call the minimum cardinality of such a set in a graph $\G$, the vertex-edge dominant edge metric dimension $\g_{emd}(\G)$ of $\G$. The new parameter $\g_{emd}$ is calculated for some common families such as paths, cycles, complete bipartite graphs, wheel and fan graphs. We also calculate $\g_{emd}$ for some Cartesian products of path with path and path with cycle. Importantly, some general results and bounds are presented for this new parameter. We also conduct a comparative analysis of $\g_{emd}$ with the dominant metric dimension of graphs. Comparison shows that these two parameters are not comparable, in general. Upon considering the class of bipartite graphs, we show that $\g_{emd}(T_n)$ of a tree $T_n$ is always less than or equal to its dominant metric dimension. However, we show that for non-tree bipartite graphs, the parameter is not comparable just like general graphs. Based on the results in this paper, we propose some open problems at the end.