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A set \( S \subseteq V \) is a dominating set of \( G \) if every vertex in \( V – S \) is adjacent to at least one vertex in \( S \). The domination number \( \gamma(G) \) of \( G \) equals the minimum cardinality of a dominating set \( S \) in \( G \); we say that such a set \( S \) is a \( \gamma \)-set.
A generalization of this is partial domination, which was introduced in 2017 by Case, Hedetniemi, Laskar, and Lipman. In \emph{partial domination}, a set \( S \) is a \( p \)-dominating set if it dominates a proportion \( p \) of the vertices in \( V \). The \( p \)-domination number \( \gamma_p(G) \) is the minimum cardinality of a \( p \)-dominating set in \( G \).
In this paper, we investigate further properties of partial dominating sets, particularly ones related to graph products and locating partial dominating sets. We also introduce the concept of a \( p \)-influencing set as the union of all \( p \)-dominating sets for a fixed \( p \) and investigate some of its properties.