Mentors: Cody Olsen (UNT graduate), Dr. Nam Trang, Dr. Stephen Jackson

Title: Cops and Robers: Monotonicity on Tunnel Cop Numbers.

Abstract: The Game of Cops and Robbers is a turn-based game with perfect information where some number of cops and a single robber are placed upon the vertices of a connected graph. The cops win if and only if on some turn a cop and the robber share a vertex; otherwise the robber wins. In our last talk, our group defined a tunnel graph as such: take an original graph and extend the distance between all vertices by adding the same number of new vertices between them. We had discussed in that talk that a tunnel graph's cop number is bounded above and below in relation to the original cop number. Recently, we showed there is monotonicity of an original graph's tunnel cop numbers on the length of the tunnels.