Problem
Research questions and friction points this paper is trying to address.
identifying codes
always-forced vertices
min-forced vertices
proper-min-forced vertices
On the Vertices That Belong to All Minimum Identifying Codes
🤖 AI Summary
本文研究了图中所有最小识别码共有的顶点问题，通过构造特定图族和理论分析给出了proper-min-forced顶点数量的上界，并证明了相关决策问题是co-NP难的。
📝 Abstract
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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
identifying codes
proper-min-forced vertices
closed-twin-free graphs
co-NP-hard
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