A Taxonomy of Perfect Domination

Abstract

A variety of terminology has been used in the literature to describe a dominating set with the property that each vertex in graph is dominated exactly once (or at most once, depending on the situation). We review the various terminologies related to perfect domination, consider some of the variations of perfect domination studied in the literature, and survey some of the main results.

Rainbow perfect domination in lattice graphs

“…literature [23,25]. We work with the following one [32] denoted with the short acronym PDS, to make a distinctive difference:…”

There is but one PDS in $\mathbb{Z}^{3}$ inducing just square components

“…The closed neighborhood of a vertex θ ∈ V in Γ is denoted N [θ]. Let [S] be the subgraph of Γ induced by S. The induced components of S, namely the connected components of [S] in Γ, are said to be the components of S. Several definitions of perfect dominating sets in graphs are considered in the literature [19,21]. We work with the following one [24] denoted with the short acronym PDS, to make a distinctive difference:…”

Perfect Domination Ratios of Archimedean Lattices

“…Perfect domination was introduced in [1] in the study of perfect codes, and has applications to facility location [21] and to efficient resource placement in a computer network [25]. A relatively recent survey of the main results in the literature on perfect domination is provided by Klostermeyer [24], which provides 57 references.…”