On the Multi-Robber Damage Number
This paper studies a variant of the “Cops and Robbers” game on graphs: $s$ robbers permanently damage vertices they visit, while a single cop aims to prevent such damage. The central question is: for which graph structures can the cop guarantee that at least three vertices remain undamaged? Employing combinatorial game theory, extremal graph theory, and constructive methods, we make three main contributions: (1) We confirm and tighten the three-vertex protection threshold for triangle-free graphs to $inom{s}{2}+2$; (2) We refute a prior conjecture for general graphs and precisely determine the minimum maximum-degree threshold for three-vertex protection within $[2inom{s}{2}-3,, 2inom{s}{2}+1]$; (3) We initiate the first systematic analysis of the two-cops versus two-robbers setting, deriving an exact formula for the number of damaged vertices on cycles and establishing an additive constant error bound on paths.