Multiset Dimensions of Trees

Abstract

Let G be a connected graph and W be a set of vertices of G. The representation multiset of a vertex v with respect to W , rm(v|W ), is defined as a multiset of distances between v and the vertices in W . If rm(u|W ) = rm(v|W ) for every pair of distinct vertices u and v, then W is called an m-resolving set of G. If G has an m-resolving set, then the cardinality of a smallest m-resolving set is called the multiset dimension of G, denoted by md(G); otherwise, we say that md(G) = ∞. In this paper, we show that for a tree T of diameter at least 2, if md(T ) < ∞, then md(T ) ≤ n − 2. We conjecture that this bound is not sharp in general and propose a sharp upper bound. We shall also provide necessary and sufficient conditions for caterpillars and lobsters having finite multiset dimension. Our results partially settled a conjecture and an open problem proposed in [4].

A New Technique to Uniquely Identify the Edges of a Graph

“…On to the contrary, if Γ does not possess a multiset resolving set, we agreed that md(Γ) = ∞. For some partial information on graphs satisfying this property, we suggest [33].…”