Local multiset dimension of comb product of tree graphs

AIMS Mathematics

article

Ridho Alfarisi¹  Liliek Susilowati²  Dafik³ 

[1] Department of Elementary School Teacher Education, University of Jember;Department of Mathematics, Universitas Airlangga;Department of Mathematics Education, University of Jember

关键词: local m-resolving set;    local multiset dimension;    comb product;    tree;    cycle;   

DOI  :  10.3934/math.2023421

学科分类：地球科学(综合)

来源: AIMS Press

【 摘 要 】

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.

【 授权许可】

CC BY