属于所有最小标识码的顶点
On the Vertices That Belong to All Minimum Identifying Codes
中文总结 AI 辅助
本文研究图中属于所有最小标识码的顶点，区分始终强制与最小强制顶点，证明真最小强制顶点数量的上界为2n/3，构造达到该界减一的图族，并证明相关判定问题是co-NP难的。
中文摘要 AI 辅助
自1998年Karpovsky、Chakrabarty和Levitin引入标识码以来，图中的标识码已被广泛研究。本文考虑图中属于每个最小标识码的顶点。此类顶点有两种类型：\u201c始终强制\u201d顶点，即属于所有标识码（无论是否为最小）的顶点；以及\u201c最小强制\u201d顶点，即属于所有最小标识码的顶点。若一个顶点是最小强制但非始终强制，则称为\u201c真最小强制\u201d顶点。我们证明了在阶数为$n$的无闭孪生图中，此类真最小强制顶点的数量上界为$2n/3$。此外，对于能被3整除的整数$n$，我们构造了一个无限图族，其中包含$2n/3-1$个此类顶点。另外，我们确定了在偶阶图中，当图包含真最小强制顶点时，边数的最大值。我们还证明了判断图中给定顶点是否为真最小强制顶点的决策问题是co-NP难的。
英文摘要
Identifying codes in graphs have been widely studied since their introduction by Karpovsky, Chakrabarty and Levitin in 1998. In this paper, we consider the vertices that are in every minimum identifying code in a graph. There are two types of such vertices: \emph{always-forced} vertices that belong to all identifying codes (minimum or not) and \emph{min-forced} vertices that belong to all minimum identifying codes. A vertex is called \emph{proper-min-forced} if it is min-forced but not always-forced. We show an upper bound $2n/3$ for the number of such proper-min-forced vertices in a closed-twin-free graph of order $n$. Moreover, for integers $n$ divisible by three, we construct an infinite family of graphs in which there are $2n/3-1$ such vertices. In addition, we determine the maximum number of edges in a graph of even order such that the graph contains proper-min-forced vertices. We also show that the decision problem of determining whether a given vertex in a graph is proper-min-forced is co-NP-hard.
