K-tuple domination on the Bishop's graph

Abstract

For a graph G = (V, E) a set 5 is a k-tuple dominating set if every vertex in V is dominated at least fc times. The minimum number of bishops needed so that every square on an nxn board is dominated k times is the k-tuple domination number, denoted γ×k(B_n). In this paper the k-tuple domination number is reduced to two possible values on the bishop's graph for k ≤ |n/2| and odd n, and is bounded between nk - k and nk for k ≤ n/2 and even n. Also found is the ×(n - 1)-domination number on the bishop's graph.