On the Multi-Robber Damage Number

We study a variant of the Cops and Robbers game on graphs in which the robbers damage the visited vertices, aiming to maximize the number of damaged vertices. For that game with one cop against $s$ robbers a conjecture was made by Carlson, Halloran and Reinhart that the cop can save three vertices from being damaged as soon as the maximum degree of the base graph is at least $\binom{s}{2} + 2$. We are able to verify the conjecture and prove that it is tight once we add the assumption that the base graph is triangle free. We also study the game without that assumption, disproving the conjecture in full generality and further attempting to locate the smallest maximum degree of a base graph which guarantees that the cop can save three vertices against $s$ robbers. We show that this number is between $2\binom{s}{2} - 3$ and $2\binom{s}{2} + 1$. Furthermore, after the game has been previously studied with one cop and multiple robbers, as well as with one robber and multiple cops, we initiate the study of the game with two cops and two robbers. In the case when the base graph is a cycle we determine the exact number of damaged vertices. Additionally, when the base graph is a path we provide bounds that differ by an additive constant.