In the literature, several identification problems in graphs have been studied, of which, the most widely studied are the ones based on dominating sets as a tool of identification. Hereby, the objective is to separate any two vertices of a graph by their unique neighborhoods in a suitably chosen dominating or total-dominating set. Such a (total-)dominating set endowed with a separation property is often referred to as a code of the graph. In this paper, we study the four separation properties location, closed-separation, open-separation and full-separation. We address the complexity of finding minimum separating sets in a graph and study the interplay of these separation properties with several codes (establishing a particularly close relation between separation and codes based on domination) as well as the interplay of separation and complementation (showing that location and full-separation are the same on a graph and its complement, whereas closed-separation in a graph corresponds to open-separation in its complement).

翻译：在文献中，图上的多种识别问题已被研究，其中基于支配集作为识别工具的问题研究最为广泛。其目标是通过在适当选择的支配集或全支配集中，利用顶点邻域的唯一性来分离图中的任意两个顶点。这种具有分离性质的（全）支配集通常被称为图的编码。本文研究了四种分离性质：定位、闭分离、开分离和全分离。我们探讨了在图中寻找最小分离集的复杂度，并研究了这些分离性质与多种编码之间的相互作用（特别建立了分离与基于支配的编码之间的紧密联系），以及分离与补图之间的相互作用（表明定位和全分离性质在图及其补图上相同，而图中的闭分离对应于其补图中的开分离）。