Distance restricted optimal pebbling in cycles

Abstract

A new variant of pebbling, “(d,t)-pebbling”, was first proposed by Chang and Shiue in an article entitled by ”An Investigation of the Game of Defend the Island”. This type of pebbling is distance restricted. That is, the distance between any start vertex and a target vertex is at most d when we apply pebbling moves. It is easy to verify that the optimal (1,1)-pebbling number is equal to the Roman domination number for any graph. Hence, it is interesting to study the (d,t)-pebbling in graphs. In this article, we first obtain a lower bound of the optimal (d,t)-pebbling number for all regular graphs when d=1,2 and then we determine the exact value of the optimal (1,t)-pebbling number for all cycles and each positive integer t.