The complement-edge dimension of corona graphs

In this study, a new concept is introduced, namely the complement-edge dimension. This concept is formed from the combination of edge metric dimension and complement metric dimension concepts. Given a connected graph G = (V(G), E(G)) with V(G) = {v1, v2,..., vn }, and an ordered set Q = {q1,..., qk} ⊆ V (G). The representation of the edge e = xy ∈ E (G) with respect to Q can be written as r (e│Q) = (d(e, qi)) = (min{d(x, qi), d (y, qi)}) with explanation d (x, qi) and d (y, qi) state the distance from vertex x to qi and from vertex y to qi for i ∈ {l,2,, k }, respectively. If there are at least two edges in G that have the same representation with respect to Q, that is the definition of a complement-edge generator set of G. The maximum cardinality of set Q is called a complement-edge basis. Further, the number of vertices of the complement-edge basis Q is called a complement-edge dimension of G, known as edim¯ (G). As a main result, the complement-edge dimension that is implemented in corona graphs between a connected graph G and any graph H is found, denoted edim¯(G⊙H). In addition, we also found the upper and lower bound of the edim¯ (G).