The \(k\)-size edge metric dimension of graphs (Q1983360)

Summary: In this paper, a new concept \(k\)-size edge resolving set for a connected graph \(G\) in the context of resolvability of graphs is defined. Some properties and realizable results on \(k\)-size edge resolvability of graphs are studied. The existence of this new parameter in different graphs is investigated, and the \(k\)-size edge metric dimension of path, cycle, and complete bipartite graph is computed. It is shown that these families have unbounded \(k\)-size edge metric dimension. Furthermore, the \(k\)-size edge metric dimension of the graphs \(P_m\,\square\,P_n\), \(P_m\,\square\,C_n\) for \(m, n \geq 3\) and the generalized Petersen graph is determined. It is shown that these families of graphs have constant \(k\)-size edge metric dimension.