本文研究了求图的\emph{geodetic number}的计算复杂性。一个图$G$的顶点集$S$是一个\emph{geodetic set}，如果$G$的任何顶点位于来自$S$的某对顶点之间的某个最短路径上。\textsc{Minimum Geodetic Set (MGS)}问题是寻找一个具有最小基数的测地集。在本文中，我们证明了解决\textsc{MGS}问题是NP-困难的最大度为6的平面图和线图。我们还表明，除非$P=NP$，没有多项式时间算法来解决\textsc{MGS}问题的次对数逼近因子（在顶点数方面），即使在图直径$2$。在积极的方面，我们给出了$n$阶图上的\textsc{MGS}问题的一个$O\left(\sqrt[3]{n}\log n\right)$ -近似算法。对于平面图的一个子类--实格图族，我们给出了求解\textsc{MGS}问题的$3$ -近似算法。
英文摘要：
In this paper, we study the computational complexity of finding the \emph{geodetic number} of graphs. A set of vertices $S$ of a graph $G$ is a \emph{geodetic set} if any vertex of $G$ lies in some shortest path between some pair of vertices from $S$. The \textsc{Minimum Geodetic Set (MGS)} problem is to find a geodetic set with minimum cardinality. In this paper, we prove that solving the \textsc{MGS} problem is NP-hard on planar graphs with a maximum degree six and line graphs. We also show that unless $P=NP$, there is no polynomial time algorithm to solve the \textsc{MGS} problem with sublogarithmic approximation factor (in terms of the number of vertices) even on graphs with diameter $2$. On the positive side, we give an $O\left(\sqrt[3]{n}\log n\right)$-approximation algorithm for the \textsc{MGS} problem on general graphs of order $n$. We also give a $3$-approximation algorithm for the \textsc{MGS} problem on the family of solid grid graphs which is a subclass of planar graphs.
Algorithms and Hardness for Geodetic Set on Tree-like Digraphs
Université Clermont Auvergne, CNRS, Clermont Auvergne INP, Mines Saint-Étienne, LIMOS（克莱蒙奥弗涅大学、法国国家科学研究中心、克莱蒙奥弗涅理工大学、圣艾蒂安国立高等矿业学校、信息与系统优化实验室）；School of Mathematics, Statistics, and Computer Science, University of Tehran（德黑兰大学数学、统计与计算机科学学院）；Indian Institute of Science Education and Research Pune（浦那科学教育与研究所）
Algorithms and complexity for geodetic sets on planar and chordal graphs
School of Computing, University of Leeds（利兹大学计算学院）；Ben-Gurion University of the Negev（内盖夫本古里安大学）；Indian Institute of Technology, Kharagpur（印度理工学院卡拉格普尔分校）
The Closed Geodetic Game: algorithms and strategies
Université Clermont Auvergne（克莱蒙奥弗涅大学）；CNRS（法国国家科学研究中心）；Mines de Saint-Étienne（圣艾蒂安国立高等矿业学校）；Clermont-Auvergne-INP（克莱蒙奥弗涅理工学院）；LIMOS（极限与优化实验室）；INRAE（法国国家农业、食品和环境研究所）；Laboratoire G-SCOP（G-SCOP实验室）；CNRS, Univ. Grenoble Alpes（法国国家科学研究中心，格勒诺布尔阿尔卑斯大学）；Ben-Gurion University of the Negev（内盖夫本古里安大学）；Indian Institute of Technology Dharwad（印度理工学院达瓦德分校）；Polytechnic University (Kyaing Tong)（桂开工科大学）
Hardness of Burning Number Problem on Regular Graphs
Indian Institute of Science Education and Research Thiruvananthapuram（印度科学教育研究所特里凡得琅分校）；Indian Institute of Science, Bengaluru（印度科学学院班加罗尔分校）；Manipal Academy of Higher Education（马尼帕尔高等教育学院）；Indian Institute of Technology Delhi（德里印度理工学院）；Kyoto University（京都大学）
Balanced spanning forests of graphs: Hardness and solving methods
KU Leuven（荷语鲁汶大学）；Research Centre for Operations Research and Business Statistics (ORSTAT)（运筹学与商业统计研究中心）；Institute of Computing, University of Campinas（坎皮纳斯大学计算研究所）
