Abstract
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 Pm □ Pn, Pm □ Cn for m, n ≥ 3 and the generalized Petersen graph is determined. It is shown that these families of graphs have constant k -size edge metric dimension.
Highlights
Kelenc et al [1] recently defined the concept of edge resolvability in graphs and initiated the study of its mathematical properties. e edge metric dimension of graph G is the minimum cardinality of edge resolving set, say X, and is denoted as βe(G)
An edge metric generator for G of cardinality βe(G) is an edge metric basis for G [1]. is concept of an edge metric generator may have a weakness with respect to possible uniqueness of the edge identifying a pair of different vertices of the graph
A set of vertices X is an edge metric generator for a graph G, whenever every two edges of G are resolved by some vertex of X. e edge metric dimension of graph G is the minimum cardinality of set X and is denoted as βe(G)
Summary
Kelenc et al [1] recently defined the concept of edge resolvability in graphs and initiated the study of its mathematical properties. e edge metric dimension of graph G is the minimum cardinality of edge resolving set, say X, and is denoted as βe(G). E edge metric dimension of graph G is the minimum cardinality of edge resolving set, say X, and is denoted as βe(G). Is concept of an edge metric generator may have a weakness with respect to possible uniqueness of the edge identifying a pair of different vertices of the graph. A set of vertices X is an edge metric generator for a graph G, whenever every two edges of G are resolved by some vertex of X. e edge metric dimension of graph G is the minimum cardinality of set X and is denoted as βe(G). E k-size edge metric dimension of G, denoted by βkse(G), is the minimum cardinality of a k-size edge resolving set of G. We present some realizable result on k-size edge metric dimension in graphs for k 1, 2
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