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识别树与给定最大度的无4-环图中的开放码

Identifying open codes in trees and 4-cycle-free graphs of given maximum degree

Centrale Méditerranée（地中海中央理工学院）；Laboratoire d’Informatique et de Systèmes, UMR 7020（信息与系统实验室，联合研究单位7020）

Dipayan Chakraborty, Florent Foucaud, Michael A. Henning

发表信息：
Discrete Applied Mathematics 386: 319-333 (2026)
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AI总结：

本研究针对最大度不超过固定值Δ≥3的无4-环无开放孪生点连通图，给出了识别开放码最小规模的上界公式，指出唯一例外图类，并构造紧例证明上界最优。

摘要：

图$G$的识别开放码是一个顶点集$S$，它同时是分离开放码（即对$G$中任意两个不同的顶点$u$和$v$，都有$N_G(u) \cap S \ne N_G(v) \cap S$）和全控制集（即对$G$中所有顶点$v$，都有$N(v) \cap S \ne \emptyset$）。当且仅当图$G$是无开放孪生点且无孤立点时，这样的集合存在；在无开放孪生点且无孤立点的图$G$中，识别开放码的最小基数记为$γ^{\rm {\small IOC}}(G)$。\n 我们研究了图的识别开放码的最小规模，及其与图的阶数和最大度的关系。对于固定的整数$\Delta \ge 3$，若$G$是阶数$n \ge 5$的连通图，不含4-环，无开放孪生点，且最大度不超过$\Delta$，则我们证明$γ^{\rm {\small IOC}}(G) \le \left( \frac{2Δ- 1}Δ \right) n$，除非$G$是通过对星图$K_{1,Δ}$的每条边恰好细分一次得到的图。\n 此外，我们通过构造达到该上界的图，证明了这个界是紧的。

英文摘要：

An identifying open code of a graph $G$ is a set $S$ of vertices that is both a separating open code (that is, $N_G(u) \cap S \ne N_G(v) \cap S$ for all distinct vertices $u$ and $v$ in $G$) and a total dominating set (that is, $N(v) \cap S \ne \emptyset$ for all vertices $v$ in $G$). Such a set exists if and only if the graph $G$ is open twin-free and isolate-free; and the minimum cardinality of an identifying open code in an open twin-free and isolate-free graph $G$, its identifying open code number, is denoted by $γ^{\rm {\small IOC}}(G)$. We study the identifying open code number of a graph, in relation with its order and its maximum degree. For $Δ$ a fixed integer at least $3$, we show that if $G$ is a connected graph of order $n \ge 5$ that contains no $4$-cycle and is open twin-free with maximum degree bounded above by $Δ$, then $γ^{\rm {\small IOC}}(G) \le \left( \frac{2Δ- 1}{2Δ} \right) n$, unless $G$ is obtained from a star $K_{1,Δ}$ by subdividing every edge exactly once. Moreover, we show that the bound is best possible by constructing graphs that reach the bound when $Δ=3$, and nearly best possible by another construction when $Δ\geq 4$, with identifying open code numbers $\left(\frac{2Δ- 4}{2Δ-3} \right) n$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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