Local multiset dimension of comb product of tree graphs

Abstract

Resolving set has several applications in the fields of science, engineering, and computer science. One application of the resolving set problem includes navigation robots, chemical structures, and supply chain management. Suppose the set $ W = \left\{{s}_{1}, {s}_{2}, \dots , {s}_{k}\right\}\subset V\left(G\right) $, the vertex representations of $ x\in V\left(G\right) $ is $ {r}_{m}\left(x\right|W) = \{d(x, {s}_{1}), d(x, {s}_{2}), \dots , 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}\left(v|W\right)\ne {r}_{m}\left(u\right|W) $ for $ uv\in E\left(G\right) $. 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 $ m{d}_{l}\left(G\right) $. In our paper, we determined the bounds of the local multiset dimension of the comb product of tree graphs.

Keywords

• local m-resolving set
• local multiset dimension
• comb product
• tree
• cycle