  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.
