Total and Double Total Domination on Octagonal Grid

Abstract

A k-total dominating set is a set of vertices such that all vertices in the graph, including the vertices in the dominating set themselves, have at least k neighbors in the dominating set. The k-total domination number γkt(G) is the cardinality of the smallest k-total dominating set. For k=1,2, the k-total dominating number is called the total and the double total dominating number, respectively. In this paper, we determine the exact values for the total domination number on a linear and on a double octagonal chain and an upper bound for the total domination number on a triple octagonal chain. Furthermore, we determine the exact values for the double total domination number on a linear and on a double octagonal chain and an upper bound for the double total domination number on a triple octagonal chain and on an octagonal grid Om,n,m≥3,n≥3. As each vertex in the octagonal system is either of degree two or of degree three, there is no k-total domination for k≥3.

Keywords

• total domination number
• double total domination number
• linear octagonal chain
• double octagonal chain
• triple octagonal chain
• octagonal grid