On the local multiset dimension ofm-shadow graph
Robiatul Adawiyah, Dafik Dafik, Ika Hesti Agustin, Rafiantika Megahnia Prihandini, Ridho Alfarisi, Ermita Rizki Albirri · Journal of Physics Conference Series · 2019
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 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 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 µ 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 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.