AI 中文总结
该研究确定了图中属于所有度量基的基强制顶点的数量上界，回答了Bagheri等人的问题，并刻画了满足特定条件的极端图类。
AI 中文摘要
图的度量基自20世纪70年代由Slater以及独立的Harary和Melter引入以来已被广泛研究。本文聚焦于图G中属于G的所有度量基的顶点的存在性，将这些顶点称为基强制顶点，并用bf(G)表示其数量。我们证明，对于阶为n、每个度量基含k个顶点的任意连通非平凡图G，有bf(G)≤2/3(n−k−1)，且该界是可达的。此外，由该结果可推导出针对任意非平凡连通图G、仅基于图阶n的界bf(G)≤2/5(n−1)，这一结果回答了Bagheri等人2016年提出的问题。我们还在上述界内给出了满足参数n、dim(G)和bf(G)≥1的完整实现，研究了图中与基强制顶点相关的一些极端情况，特别给出了满足bf(G)=2且dim(G)=n−4的图的完整刻画。
英文摘要
Metric bases of graphs have been widely studied since their introduction in the 1970's by Slater and, independently, by Harary and Melter. In this paper, we concentrate on the existence of vertices in a graph $G$ that belong to all metric bases of $G$. We call these basis forced vertices, and denote the number of them by $\mathrm{bf}(G)$. We show that $\mathrm{bf}(G)\le 2/3(n-k-1)$ for any connected nontrivial graph $G$ of order $n$ having $k$ vertices in each metric basis. In addition, we show that this bound can be attained. Furthermore, the previous result implies the bound $\mathrm{bf}(G)\le 2/5(n-1)$ formulated in terms of the order $n$ of the graph for any nontrivial connected graph $G$. This result answers a question posed by Bagheri et al. in 2016. Moreover, we provide a complete realization of the parameters $n$, $\dim(G)$ and $\mathrm{bf}(G) \ge 1$ within the previous bounds. We consider some extremal cases related to basis forced vertices in a graph, in particular, we give a full characterization of the graphs with $\mathrm{bf}(G) = 2$ and $\dim(G) = n-4$.
Comments19 pages, 8 figures. Parts of this paper appeared in CALDAM 2025
