关于有向图上识别码的研究
On identifying codes on oriented graphs
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中文总结 AI 辅助
本文从计算复杂性视角研究有向图上的识别码，针对d-正则图族的F-Id Code问题，证明当d≤1时该问题多项式时间可解，d≥2时为NP完全问题，建立了该问题的完全二分性。
中文摘要 AI 辅助
本文从计算复杂性视角研究有向图中的识别码。我们探讨$\boldsymbol{\textit{F}}$-Id Code问题，该问题以简单图$G$和顶点子集$C$（$C$在$\boldsymbol{\textit{F}}$族中诱导出一个子图）为输入，要求判断是否能对$G$进行定向，使得$C$成为其有向识别码。针对$d$-正则图族$\boldsymbol{\textit{F}}_d$，我们建立了完全二分性：证明当$d \boldsymbol{\textit{F}}_d$时该问题是多项式时间可解的，而当$d \boldsymbol{\textit{F}}_d$时该问题是NP完全的。
英文摘要
This article studies identifying codes in oriented graphs from a computational complexity perspective. We investigate the $\mathcal{F}$-Id Code problem, where given a simple graph $G$ and a vertex subset $C$, which induces a subgraph in the family $\mathcal{F}$, as inputs and ask whether it is possible to orient $G$ in such a way that $C$ becomes its oriented identifying code. Focusing on the family $\mathcal{F}_d$ of $d$-regular graphs, we establish a complete dichotomy by proving that the problem is polynomial-time solvable for $d\leq1$ and NP-complete for all $d\geq2$.
