The Odd Domination Number of a Graph

Let \(G = (V, E)\) be a simple, undirected graph. A set of vertices \(D\) is called an odd dominating set if for every vertex \(v \in V(G)\), \(|N[v] \cap D| \equiv 1 \pmod{2}\). The minimum cardinality of an odd dominating set is called the odd domination number of \(G\). It is well known that every graph contains an odd dominating set, but this parameter has been studied very little. Our aim in this paper is to explore some basic features of the odd domination number and to compare it with the domination number of the graph, denoted by \(\gamma(G)\). In addition, extremal values of \(\gamma_{odd}(G)\) are calculated for several classes of graphs and a Nordhaus-Gaddum type inequality \(\gamma_{odd}(G) + \gamma_{odd}(\overline{G})\) is considered.