  Domination problems in general can capture situations in which some entities
have an effect on other entities (and sometimes on themselves). The usual goal
is to select a minimum number of entities that can influence a target group of
entities or to influence a maximum number of target entities with a certain
number of available influencers. In this work, we focus on the distinction
between \textit{internal} and \textit{external} domination in the respective
maximization problem. In particular, a dominator can dominate its entire
neighborhood in a graph, internally dominating itself, while those of its
neighbors which are not dominators themselves are externally dominated. We
study the problem of maximizing the external domination that a given number of
dominators can yield and we present a 0.5307-approximation algorithm for this
problem. Moreover, our methods provide a framework for approximating a number
of problems that can be cast in terms of external domination. In particular, we
observe that an interesting interpretation of the maximum coverage problem can
capture a new problem in elections, in which we want to maximize the number of
\textit{externally represented} voters. We study this problem in two different
settings, namely Non-Secrecy and Rational-Candidate, and provide
approximability analysis for two alternative approaches; our analysis reveals,
among other contributions, that an earlier resource allocation algorithm is, in
fact, a 0.462-approximation algorithm for maximum external domination in
directed graphs.
