### Total domination in digraphs

#### Abstract

A vertex subset* S* of a digraph* D* is called a dominating set of *D* if every vertex not in *S* is adjacent from at least one vertex in *S*. A dominating set *S* of *D* is called a total dominating set of *D* if the subdigraph of *D* induced by *S* has no isolated vertices. The total domination number of *D*, denoted by *γt(D)*, is the minimum cardinality of a total dominating set of *D*. We show that if *D* is a rooted tree, a connected contrafunctional digraph or a strongly connected digraph of order *n* ≥ 2, then* γt(D)* ≤ 2(*n* + 1)/3 and if *D* is a digraph of order n with minimum in-degree at least one whose connected components are isomorphic to neither* c̅ _{2}* nor

*c̅*, then

_{5}*γt(D)*≤ 3

*n*/4, where

*c̅*and

_{2}*c̅*denote the directed cycles of order 2 and 5 respectively. Moreover, we characterize the corresponding digraphs achieving these upper bounds.

_{5}**Keywords:** Total domination, rooted tree, contrafunctional digraph, directed graph

http://dx.doi.org/10.2989/16073606.2017.1288664