On full-separating sets and related codes in graphs

A domination-based identification problem on a graph $G$ is one where the objective is to choose a subset $C$ of the vertex set of $G$ such that $C$ has both, a domination property, that is, $C$ is either a dominating or a total-dominating set of $G$, and a separation property, that is, any two distinct vertices of $G$ must have distinct closed or open neighborhoods in $C$. Such a set $C$ is often referred to as a code in the literature of identification problems. In this article, we introduce a new separation property, called full-separation, as it combines aspects of the two well-studied properties of closed- and open-separation. We study it in combination with both domination and total-domination and call the resulting codes full-separating dominating codes (or FD-codes for short) and full-separating total-dominating codes (or FTD-codes for short), respectively. Incidentally, FTD-codes have also been introduced in the literature of identification problems under the name of strongly identifying codes (or SID-codes for short) and under a differently formulated definition. In this paper, we address the conditions for the existence of FD- and FTD-codes, bounds for their size, their relation to codes of the other types and present some extremal cases for these bounds and relations. We further show that the problems of determining an FD- or an FTD-code of minimum cardinality in a graph are NP-hard. We also show that the cardinalities of minimum FD- and FTD-codes of any graph differ by at most one, but that it is NP-hard to decide whether or not they are equal for a given graph in general.