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 $W(n,k)$.

中文翻译：

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以公制为基础，轮辐连续丢失

如果$ G $为连通图，则两个顶点$ u，v之间的$ distance $ $ d（u，v）$是V（G）$中最短路径的长度。假设$ W = \ {w_1，w_2，\ dots，w_k \} $是$ G $的有序顶点集合，而$ v $是$ G $的顶点。$ v $相对于$ W $的$ representation $ $ r（v | W）$是k元组$（d（v，w_1），d（v，w_2），\ dots，d（v， w_k））$。如果$ G $的每个顶点由其与$ W $顶点的距离唯一地标识，或者等效地如果$ G $的不同顶点具有不同的表示形式，则$ W $称为$解析集$或$定位集$。关于$ W $。最小基数的解析集称为$ G $的$ metric基数$，此基数是$ G $的$ metric维数$，用$ \ beta（G）$表示。Siddiqui和Imran最近研究了一些车轮相关图的度量尺寸。在本文中，我们研究了连续丢失$ k $个辐条的车轮的度量尺寸，用$ W（n，k）$表示。我们计算了度量维数$ W（n，k）$的精确值，该值表明轮辐连续丢失的车轮具有不受限制的度量维数。要求对度量维度无限制的图形进行表征是很自然的。本文还研究了求解一组$ W（n，k）$的交换性质，结果表明，向量空间中基的交换性质不适用于$ k $的最小车轮解集。 -以$ W（n，k）$表示的连续缺失辐条。要求对度量维度无限制的图形进行表征是很自然的。本文还研究了求解一组$ W（n，k）$的交换性质，结果表明，向量空间中基的交换性质不适用于$ k $的最小车轮解集。 -以$ W（n，k）$表示的连续缺失辐条。要求对度量维度无限制的图形进行表征是很自然的。本文还研究了求解一组$ W（n，k）$的交换性质，结果表明，向量空间中基的交换性质不适用于$ k $的最小车轮解集。 -以$ W（n，k）$表示的连续缺失辐条。