A note on total domination

Abstract

For a graph G, let γ(G), γ1(G), i(G) and ir(G) denote the domination, total domination, independent domination and irredundance numbers of G, respectively. The following conjectures due to Robyn Dawes are proved: 1. (i) γ(G) + γ1(G) ≤ p and 2. (ii) i(G) + γ1(G) ≤ p, where |V(G)| = p > 2. It is also shown that 1. (iii) γ1(G) ≤ 2ir(G) and 2. (iv) γ(G) ≤ 2ir(G) - (k + 1). where k is the maximum number of isolates in an ir(G) set. This last result improves the result, obtained independently by Bollóbas and Cockayne [6], Allan and Laskar [2]. © 1984.