The complement edge metric dimension of graphs

In this paper, we construct the new concept namely the complement edge metric dimension on the graph, which is the result of combining two concepts. The first concept is edge metric dimension and in the second concept is complement metric dimension. Let given a graph G=(V(G), E(G)) where V(G) = {v1, v2, ..., vn} and also a set M = {m1, ..., mk} ⊂ V(G) that indicate connected graph and an ordered set, respectively. The representation of the edge e=xy∈E(G) with respect to M can be written as r(e|M) = (d(e, mi))=(min{d(x, mi), d(y, mi)}) where d(x, mi) and d(y, mi) indicate the distance from vertex x to mi and from vertex y to mi for i∈{1,2, ..., k}, respectively. The definition of a complement edge resolving set of G is if any two distinct edges in G have the same representation with respect to M. If set M has maximum cardinality then it is called a complement edge basis. Next, the number of vertices of M is called the complement edge metric dimension of G that can be written as edim¯(G). As a result, the complement edge metric dimension which is implemented in basic graphs, namely graph Pn, graph Sn, graph Cn, and graph Kn will be determined.