Tight Bounds for the Split Domination Number of a Nearly Regular Tournament

A set of vertices, \( S \), in a digraph \( D \), is split dominating provided it is:

1. dominating and
2. \( D[V(D) \setminus S] \) is either trivial or has a lower level of connection than \( D \).

In this paper, we consider split dominating sets in strongly connected tournaments. The split domination number of a strongly connected tournament \( T \), denoted by \( \gamma_s(T) \), is the minimum cardinality of a split dominating set for that tournament.

The authors previously gave a tight lower bound for \( \gamma_s(T) \) when \( T \) is regular. In this paper, we show that when \( T \) is a nearly regular \( 2k \)-tournament, then \( \gamma_s(T) \geq \lceil \frac{2k}{3} \rceil \) and this bound is tight.