Improving a Nordhaus–Gaddum type bound for total domination using an algorithm involving vertex disjoint stars

Abstract

A Nordhaus–Gaddum-type result is a (tight) lower or upper bound on the sum or product of a parameter of a graph and its complement. In Henning et al. (2011) the authors (Henning et al.) show that if G1⊕G2=K(s,s), and neither G1 nor G2 has isolated vertices, then the product γt(G1)γt(G2) is at most max{8s,⌊(s+6)2∕4⌋}, where γt is the total domination number. In this paper we will use a vertex disjoint star covering technique, to significantly improve the mentioned bound. In particular, we will show that if G1⊕G2=K(s,s), and neither G1 nor G2 has isolated vertices, then γt(G1)γt(G2)≤max{8s,⌊(s+5)2∕4⌋}.