Domination and Independence on the Rectangular Torus by Rooks and Bishops

A set S V is a dominating set of a graph G = (V; E) if each vertex in V is either in S or is adjacent to a vertex in S. A vertex is said to dominate itself and all its neighbors. The domination number (G) is the minimum cardinality of a dominating set of G. A set S V is an independent set of vertices if no two vertices in S are adjacent. The independence number, B0 (G), is the maximum cardinality of an independent set of G. Both (G) and B0 (G) are pieces of the six part domination chain: ir (G) (G) i (G) B0 (G) (G) IR (G). Watkins has computed the domination numbers of rooks and bishops on the square torus. In this paper we compute the domination, total domination, independent domination and independence numbers of the bishop and rook on the rectangular m n toroidal board.

• Type: Journal
• Year: 2009
• Where: FCS
• Authors: Joe DeMaio, William Faust
• Keywords: Computer Science, Dominating Set, Domination Numbers, FCS 2009, Independence Numbers