Patrolling cop vs omniscient robber

We study a variant of the classical Cops and Robbers game with one cop and one robber. The cop follows a fixed walk on the graph, called a patrol, that is chosen before the game begins. The robber is omniscient and knows the entire patrol in advance. A capture occurs when the robber comes within a given distance of the cop, and this distance is referred to as the capture distance.
The patrol capture radius, $\tilde{\rho}{(G)}$, is the minimum radius of capture required for the cop to always be able to capture the robber on a connected graph $G$, under optimal play. We initiate a systematic study of this parameter for several graph classes. We determine the exact value of $\tilde{\rho}{(G)}$ for trees, establish upper and lower bounds for grids, and analyze the parameter for various families of chordal graphs, including interval graphs and caterpillars. Along the way, we develop general tools and structural results that may be of independent interest for the study of pursuit-evasion games with predetermined patrols and limited information.