On the adjacency dimension of some star related trees

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Document Type : Original Article

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Department of Mathematics, Faculty of Science, Imam Khomeini International University, Qazvin, Iran

Abstract

‎Locating or resolving sets are introduced as a graph-theoretic model of robot navigation and has different applications in diverse areas like network discovery‎, ‎computer science and chemistry‎. ‎These applications leads to some graph parameters‎, ‎like the metric dimension and the adjacency dimension‎.‎ A subset $S$ of the vertices of a graph ‎$‎G‎$‎ is an adjacency resolving set for $G$ if for each pair of distinct vertices‎ ‎$x‎, ‎y \in V(G)\setminus S$, there exists $s \in S$ which is adjacent to exactly one of these two vertices‎. ‎An adjacency resolving set with the minimum cardinality is called an adjacency basis and its cardinality is the adjacency dimension of $G$. ‎Since the problem of computing the adjacency dimension of a graph is NP-hard‎, ‎finding the adjacency dimension of special classes of graphs or obtaining good bounds on this invariant is valuable‎. ‎In this paper we determine the adjacency dimension of some famous star related trees.

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AUT Journal of Mathematics and Computing

Articles in Press, Accepted Manuscript
Available Online from 12 January 2025

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