On the Local Multiset Dimension of Comb Product Graphs

Abstract

One of the topics of distance in graphs is resolving set problems. This topic has many application in science and technology namely the application of resolving set problems in networks is one of the describe navigation robots, chemistry structure, and computer sciences. Suppose the set $W=\{s_1,s_2,…,s_k\}\subset V(G)$, the vertex representations of $\in V(G)$ is $r_m(x|W)=\{d(x,s_1),d(x,s_2),…,d(x,s_k)\}$, where $d(x,s_i)$ is the length of the shortest path of the vertex x and the vertex in $W$ together with their multiplicity. The set W is called a local $m$-resolving set of graphs G if $r_m (v│W)\neq r_m (u|W)$ for $uv\in E(G)$. The local $m$-resolving set having minimum cardinality is called the local multiset basis and its cardinality is called the local multiset dimension of $G$, denoted by $md_l(G)$. In our paper, we determined the establish bounds of local multiset dimension of graph resulting comb product of two connected graphs.