How Many Lions Are Needed to Clear a Grid?

Abstract

We consider a pursuit-evasion problem where some lions have the task to clear a grid graph whose nodes are initially contaminated. The contamination spreads one step per time unit in each direction not blocked by a lion. A vertex is cleared from its contamination whenever a lion moves to it. Brass et al. [5] showed that n/2 lions are not enough to clear the n x n-grid. In this paper, we consider the same problem in dimension d > 2 and prove that Θ(nd-1/√d) lions are necessary and sufficient to clear the nd-grid. Furthermore, we analyze a problem variant where the lions are also allowed to jump from grid vertices to non-adjacent grid vertices.

Keywords

• lion and man
• safe path planning
• avoidance number
• pursuit games
• graph separator
• graph search
• isoperimetric inequality