On the sum of out-domination number and in-domination number of digraphs

A vertex subset S of a digraph D = (V, A) is called an out-dominating (resp., in-dominating) set of D if every vertex in V - S is adjacent from (resp., to) some vertex in S. The out-domination (resp., in-domination) number of D, denoted by gamma(+) (D) (resp., gamma(-)(D)), is the minimum cardinality of an outdominating (resp., in-dominating) set of D. In 1999, Chartrand et al. proved that gamma(+)(D) + gamma(-) (D) <= 4n/3 for every digraph D of order n with no isolated vertices. In this paper, we determine the values of gamma(+) (D) + gamma(-) (D) for rooted trees and connected contrafunctional digraphs D, based on which we show that gamma(+)(D)+ gamma(-) (D) <= (2k+2)n/(2k+1) for every digraph D of order n with minimum out-degree or in-degree no less than 1, where 2k + 1 is the length of a shortest odd directed cycle in D. Our result partially improves the result of Chartrand et al. In particular, if D contains no odd directed cycles, then gamma(+) (D) + gamma(-) (D) <= n.