On the local multiset dimension of a graph

In robot navigation, the concept of local multiset in graph theory can be used to calculate the minimum number of sensors required to uniquely identify each point in a network. By modeling every point in the network into a vertex and their connections into edges, the solution lies in determining the minimum number of reference points necessary to identify every point in the network uniquely. Based on the distances between vertices, local multiset dimension (LMD) quantifies the minimum required number of vertices to define the unique position of every vertex in the graph. By strategically placing sensors at these reference points in a given network, LMD helps the robot map its environment and determine its location based solely on its distances to these reference points. Interest in this area of graph theory has increased in recent years. In this paper, some characterizations of the local multiset basis are discussed and extended to some graphs derived from path.