On The Local Multiset Dimension of some Families of Graphs

Let G be a connected graph. G is said to be unicyclic if it contains exactly one cycle, and bicyclic if the number of edges equals the number of vertices plus one. For a k-ordered set W = {s1, s2, , sk} ⊂ V(G), the multiset representation of a vertex x in G with respect to W is given as rm(x|W) = {d(x, s1), d(x, s2), , d(x, sk)}, where d(x, si) is the distance between x and the ordered subset si of W together with their multiplicities. The set W is called a local m-resolving set of G if for every uν ∈ E(G), rm(u|W) ≠ rm(ν|W). The local m-resolving set with minimum cardinality is called the local multiset basis and its cardinality is called the local multiset dimension of G, denoted by mdl (G). If G has no local m- resolving set, we write mdt (G) = ∞ and say that G has an infinite local multiset dimension. In this paper, we determine the local multiset dimension of the unicyclic and bicyclic graphs.