Local multiset dimension of related cycle graphs

Abstract

Suppose the set W = {s1, s2,…, sk} is a subset of the vertex set V(G). The representation of a vertex v of G with respect to W as follows rm(v|W) = { d(v, s1), d(v, s2), … , d(v, sk)} where d(v, si),1 ≤ i ≤ k is the distance between the vertex v with the vertices of set W together with their multiplicities. The set W is called the m-local resolving set of G if every two adjacent vertices of G have distinct representation with respect to W. If G has an m-local resolving set, then an m-local resolving set having minimum cardinality is called a local multiset basis and its cardinality is called the local multiset dimension of G, denoted by mdl (G). We say that G has an infinite local multiset dimension and we write mdl(G) = ∞. In this paper, we determine the local multiset dimension of related cycle graphs namely kayak paddles graph, and cycles with chord.