On the local multiset dimension of m-shadow graph

Let G = (V, E) be a simple and connected graph with edge set E and vertex set V. Suppose W = {s(1), s(2), ... , s(k)} is a subset of vertex set V(G), the representation multiset of a vertex v of G with respect to W is r(m)(v vertical bar W) = {d(v, s(1)),d(v, s(2)),...,d(v, s(k))} where d(v, s(i)) is a distance between v and the vertices in W together with their multiplicities. The resolving set W is a local resolving set of G if r(m)(v vertical bar W) not equal r r(m) (u vertical bar W) for every pair u, v of adjacent vertices of a graph G. The minimum local resolving set W is a local multiset basis of G. If G has a local multiset basis, then its cardinality is called local multiset dimension, denoted by mu l(G). In this paper, we analyzed the local multiset dimension of m-shadow graph. The m shadow of a connected graph G, denoted by D-m(G), is constructed by taking m copies of G, say G(1), G(2) circle dot, G(m) then join each vertex u in G(i) to the neighbors of the corresponding vertex v in G(j), where 1 <= i and j <= m. We will investigate the characterization and exact value of local multiset dimension of m-shadowing of cycle graph, m-shadowing of star graph, m-shadowing of path graph, and m-shadowing of complete graph.