AIMS Mathematics, 2020, 5(6): 6221-6232. doi: 10.3934/math.2020400

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On the metric basis in wheels with consecutive missing spokes

1 Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan 60800, Pakistan
2 Informetrics Research Group, Ton Duc Thang University, Ho Chi Minh City, Vietnam
3 Faculty of Mathematics & Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

If $G$ is a connected graph, the $distance$ $d(u, v)$ between two vertices $u, v \in V(G)$ is the length of a shortest path between them. Let $W = \{w_1,w_2, \dots ,w_k\}$ be an ordered set of vertices of $G$ and let $v$ be a vertex of $G$. The $representation$ $r(v|W)$ of $v$ with respect to $W$ is the k-tuple $(d(v,w_1), d(v,w_2), \dots , d(v,w_k))$. $W$ is called a $resolving set$ or a $locating set$ if every vertex of $G$ is uniquely identified by its distances from the vertices of $W$, or equivalently if distinct vertices of $G$ have distinct representations with respect to $W$. A resolving set of minimum cardinality is called a $metric basis$ for $G$ and this cardinality is the $metric dimension$ of $G$, denoted by $\beta(G)$. The metric dimension of some wheel related graphs is studied recently by Siddiqui and Imran. In this paper, we study the metric dimension of wheels with $k$ consecutive missing spokes denoted by $W(n,k)$. We compute the exact value of the metric dimension of $W(n,k)$ which shows that wheels with consecutive missing spokes have unbounded metric dimensions. It is natural to ask for the characterization of graphs with an unbounded metric dimension. The exchange property for resolving a set of $W(n,k)$ has also been studied in this paper and it is shown that exchange property of the bases in a vector space does not hold for minimal resolving sets of wheels with $k$-consecutive missing spokes denoted by $W(n,k)$.
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