Camacho Campos Carlos, Camacho Moreno José Carlos, Kuziak Dorota, Raza Zahid, Yero Ismael G
Departamento de Ingeniería Civil, Universidad de Cádiz, Algeciras Campus, Spain.
Departamento de Matemáticas, Universidad de Cádiz, Algeciras Campus, Spain.
PLoS One. 2025 Jun 17;20(6):e0325565. doi: 10.1371/journal.pone.0325565. eCollection 2025.
A model for mobility of robots keeping the property of uniquely recognizing the vertices of a given network is considered in this work. This is made in order to detect failures or intruders, by means of dynamic vectors of distances to the set of mobile robots. We consider the smallest set of robots that can be placed in a set of nodes of a network that forms a resolving set, which is a structure of a graph such that it uniquely recognizes all the vertices of the graph by using distances. We are then focused on allowing such robots to move from one vertex to another adjacent one, through the edges of the whole graph. At each performed movement we require that the new set of covered nodes forms a resolving set. This process allows the robots to recognize all the vertices of the graph, independently on the position in which they are located. In this sense, the notion of mobile metric dimension is introduced in this article, and the study of its primary combinatorial properties is initiated. We relate this parameter with the classical metric dimension and the resolving number of graphs and compute its value for several graph classes.
本文考虑了一种机器人移动模型，该模型保持唯一识别给定网络顶点的特性。这样做是为了通过到移动机器人集合的动态距离向量来检测故障或入侵者。我们考虑可以放置在网络节点集合中形成分辨集的最小机器人集合，分辨集是一种图结构，它通过使用距离唯一识别图的所有顶点。然后，我们专注于允许这些机器人通过整个图的边从一个顶点移动到另一个相邻顶点。在每次执行的移动中，我们要求新覆盖的节点集形成一个分辨集。这个过程允许机器人独立于它们所处的位置识别图的所有顶点。从这个意义上说，本文引入了移动度量维数的概念，并开始研究其主要组合性质。我们将这个参数与经典度量维数和图的分辨数相关联，并计算了几个图类别的值。
