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

TL;DR

This paper introduces a new graph parameter called the vertex-edge dominant edge metric dimension, which combines dominating and resolving sets, and explores its properties, calculations for various graph families, and comparisons with existing metrics.

Contribution

The paper defines the vertex-edge dominant edge metric dimension, calculates it for several graph families, and compares it with the classical dominant metric dimension, revealing non-comparability.

Findings

Calculated $ ext{g}_{emd}$ for paths, cycles, bipartite, wheel, and fan graphs.

Established bounds and general results for $ ext{g}_{emd}$.

Compared $ ext{g}_{emd}$ with dominant metric dimension, showing non-comparability.

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.