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

Abstract

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 ${\gamma }_s(T)\ge \lceil \frac{2k}{3}\rceil$ and this bound is tight.