WSEAS Transactions on Mathematics
Print ISSN: 1109-2769, E-ISSN: 2224-2880
Volume 22, 2023
On The Local Multiset Dimension of some Families of Graphs
Authors: , , ,
Abstract: 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 $$W = {s_{1},s_{2},…,s_{k} } ⊂V(G)$$, the multiset representation of a vertex x in G with respect to W is given as $$r_{m} (x|W)={d(x,s_{1}),d(x,s_{2}),…,d(x,s_{k})}$$, where $$d(x,s_{i})$$ is the distance between x and the ordered subset $$s_{i}$$ of W together with their multiplicities. The set W is called a local m-resolving set of G if for every $$uv∈E(G), r_m (u|W)≠ r_m (v|W)$$. 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 $$md_{l} (G)$$. If G has no local m-resolving set, we write $$md_{t} (G) = ∞$$ 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.
Search Articles
Keywords: Networks infrastructure, local m-resolving set, local multiset dimension, unicyclic graphs, bicyclic graphs
Pages: 64-69
DOI: 10.37394/23206.2023.22.8