An investigation of the game of Defend the Island

Abstract

Motivated by the problem of “Defend the Roman Empire”, we propose a variant of graph pebbling, named ( d , t )-pebbling, and use this idea to devise a two-player game of imperfect information, named “Defend the Island”. On an island, the target castle attacked by the opponent is securable if t Field Armies (FAs) can reach the target castle from neighboring castles within a distance of d. The aim is to obtain more points than the opponent before an insecure castle on the island is attacked. There is a minimum number of FAs initially deployed on the island so that each castle is securable. In graph pebbling, this number is called the optimal ( d , t )-pebbling number of a graph G, denoted by π ( d , t ) ∗ ( G ). When the number of FAs is less than π ( d , t ) ∗ ( G ), there exists at least one insecure castle. This property is the core idea to devise the new game. In addition, when the castles forms a cycle, that is, the graph is a cycle, we give a lower bound of π ( d , t ) ∗ ( G ) for d = 1 , 2 and determine the exact value of π ( d , t ) ∗ ( G ) for the ordered pairs ( d , t ) = ( 2 , 2 ) , ( 1 , 4 m ) and ( 2 , 10 m ), where m is any positive integer.