Abstract: In pursuit-evasion games, a set of pursuers attempts to locate, eliminate, or contain an evader in a network. The rules, specified from the outset, greatly determine the difficulty of the questions posed above. For example, the evader may be visible, but the pursuers may have limited movement speed, only moving to nearby vertices adjacent to them.
Central to pursuit-evasion games is the idea of optimizing certain parameters, whether they are the search number, burning number, or localization number, for example. We report on progress in several pursuit-evasion games on graphs and conjectures arising from their analysis. Finding the values, bounds, and algorithms to compute these graph parameters leads to topics intersecting graph theory, the probabilistic method, and geometry.