METRIC DIMENSION AND LOCALIZATION NUMBER ON SOME GRAPHS FOR COPS AND ROBBER GAME

Abstract

This paper discusses the other variant of the Cops and Robber game inspired by the problem of determining the actual location of a walking mobile user. Given a graph G as a representation of the search area of a mobile user who in this paper is analogous as the robber. On each round, the cops will receive distance information between them and the robber. The cops are said to have won the game, if they could determine the occupied vertex of the robber, if not then the robber win. Metric dimension of graph G is defined as the minimum number of cops needed so that they win the game in one round of play, while localization number of graph G is defined as the minimum number of cops needed so that the cops can win the game in one or several round. The result shows that the metric dimension and localization number of graph G are different on several graphs and it is also proved that the metric dimension of graph G will always be less than or equal to the difference between the order of graph G and its diameter and the localization number of graph G will always be less than or equal to the pathwidth of graph G. Keywords: Cops and Robber game, metric dimension, localization number