有向图的监测弧-测地集
Monitoring arc-geodetic sets of oriented graphs
AI总结：
本文将监测边-测地集概念扩展到有向图，提出监测弧-测地集，确定基本图类的相关参数，刻画特殊图类，并证明相关判定与优化问题的NP难和NP完全性。
摘要：
图中的监测边-测地集是指这样的顶点子集：图的每条边都必须位于监测集中两个顶点之间的所有最短路径上。这些对象由 Foucaud、Krishna 和 Ramasubramony Sulochana 在一项工作中引入，并与网络监测领域中诸如距离边监测等若干先前概念相关。在本工作中，我们探索将这些概念扩展到有向图，以对有向网络进行建模，并将这些对象称为监测弧-测地集。我们还定义了无向图的下监测弧-测地数和上监测弧-测地数，分别为该图所有定向的监测弧-测地数的最小值和最大值。我们确定了基本图类（如二部图、树、圈等）的监测弧-测地数。然后，我们刻画了每个监测弧-测地集都是整个顶点集的图，并刻画了竞赛图的解。我们还通过研究两个算法问题来涵盖一些复杂性方面的内容。我们证明，判断一个无向图是否存在一种定向使得最小监测弧-测地集为整个顶点集的问题是 NP-难的。我们还证明，在最大度为 4 的有向图上，寻找大小至多为 $k$ 的监测弧-测地集的问题是 $NP$-完全的。
英文摘要：
Monitoring edge-geodetic sets in a graph are subsets of vertices such that every edge of the graph must lie on all the shortest paths between two vertices of the monitoring set. These objects were introduced in a work by Foucaud, Krishna and Ramasubramony Sulochana with relation to several prior notions in the area of network monitoring like distance edge-monitoring. In this work, we explore the extension of those notions unto oriented graphs, modelling oriented networks, and call these objects monitoring arc-geodetic sets. We also define the lower and upper monitoring arc-geodetic number of an undirected graph as the minimum and maximum of the monitoring arc-geodetic number of all orientations of the graph. We determine the monitoring arc-geodetic number of fundamental graph classes such as bipartite graphs, trees, cycles, etc. Then, we characterize the graphs for which every monitoring arc-geodetic set is the entire set of vertices, and also characterize the solutions for tournaments. We also cover some complexity aspects by studying two algorithmic problems. We show that the problem of determining if an undirected graph has an orientation with the minimal monitoring arc-geodetic set being the entire set of vertices, is NP-hard. We also show that the problem of finding a monitoring arc-geodetic set of size at most $k$ is $NP$-complete when restricted to oriented graphs with maximum degree $4$.
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