Several different types of identification problems have been already studied in the literature, where the objective is to distinguish any two vertices of a graph by their unique neighborhoods in a suitably chosen dominating or total-dominating set of the graph, often referred to as a \emph{code}. To study such problems under a unifying point of view, reformulations of the already studied problems in terms of covering problems in suitably constructed hypergraphs have been provided. Analyzing these hypergraph representations, we introduce a new separation property, called \emph{full-separation}, which has not yet been considered in the literature so far. We study it in combination with both domination and total-domination, and call the resulting codes \emph{full-separating-dominating codes} (or \emph{FD-codes} for short) and \emph{full-separating-total-dominating codes} (or \emph{FTD-codes} for short), respectively. We address the conditions for the existence of FD- and FTD-codes, bounds for their size and their relation to codes of the other types. We show that the problems of determining an FD- or an FTD-code of minimum cardinality in a graph is NP-hard. We also show that the cardinalities of minimum FD- and FTD-codes differ by at most one, but that it is NP-complete to decide if they are equal for a given graph in general. We find the exact values of minimum cardinalities of the FD- and FTD-codes on some familiar graph classes like paths, cycles, half-graphs and spiders. This helps us compare the two codes with other codes on these graph families thereby exhibiting extremal cases for several lower bounds.

翻译：文献中已经研究了几种不同类型的识别问题，其目标是通过在图的一个适当选择的支配集或全支配集（通常称为\emph{码}）中顶点的唯一邻域来区分图的任意两个顶点。为了从统一的角度研究此类问题，已有研究以在适当构造的超图中的覆盖问题的形式对已研究问题进行了重新表述。通过分析这些超图表示，我们引入了一种新的分离性质，称为\emph{全分离}，该性质迄今尚未在文献中被考虑。我们结合支配和全支配来研究它，并分别将得到的码称为\emph{全分离支配码}（简称\emph{FD-码}）和\emph{全分离全支配码}（简称\emph{FTD-码}）。我们探讨了FD-码和FTD-码存在的条件、其大小的界以及它们与其他类型码的关系。我们证明了在图中确定最小基数的FD-码或FTD-码的问题是NP-难的。我们还证明了最小FD-码和FTD-码的基数最多相差一，但一般而言，对于给定图判断它们是否相等是NP-完全的。我们在一些熟悉的图类上找到了FD-码和FTD-码的最小基数的精确值，例如路径、圈、半图和蜘蛛图。这帮助我们在这类图族上将这两种码与其他码进行比较，从而为几个下界展示了极值情况。