On full-separating sets and related codes in graphs

TL;DR

This work introduces full-separation, a new graph-separation property that amalgamates closed- and open-separation, and defines FD-codes and FTD-codes as full-separating dominating and full-separating total-dominating sets. It unifies these codes within the existing X-code hypergraph framework and proves equivalences, existence criteria, and interdependencies with other X-numbers, establishing that FD/FTD are typically stronger than many classical counterparts. The authors prove NP-hardness for minimum FD/FTD-code problems and show that the two cardinalities are within one of each other, though deciding equality is itself hard. They analyze exact values for several graph families (paths, cycles, half-graphs, headless spiders), illustrating when the FD- and FTD-numbers coincide or differ, thereby highlighting both theoretical and practical implications for network monitoring and fault detection. The paper opens avenues for broader hypergraph classifications of domination/separation problems and encourages exploration of further graph families and extremal cases where FD/FTD codes achieve order-sized bounds or tight gaps to other X-numbers.

Abstract

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.

Figures (6)

• Figure 1: Minimum X-codes in the bull graph (the black vertices belong to the code), where (a) is an ID-code, (b) an ITD-code, (c) both an LD- and LTD-code, (d) both an OD- and OTD-code.
• Figure 2: Minimum X-codes in a graph (the black vertices belong to the code), where (a) is an FD-code, (b) an FTD-code.
• Figure 3: The relations between the X-numbers for all $X \in {\textsc{Codes}}$, where $X' \longrightarrow X$ stands for $\gamma^{{\textsc{X}}'}(G) \leq \gamma^{\textsc{X}}(G)$ and solid (respectively, dotted and dashed) arrows refer to case (a) (respectively, (b) and (c)) of Corollary \ref{['cor_all_relations']}.
• Figure 4: Polynomial-time construction of the graph $G^\psi$ from an instance $\psi = (X,\mathcal{C})$ of 3-SAT as in Reduction \ref{['red_NP']}. The black vertices in (c) represent those in a code described in Lemma \ref{['aplem_NP_1']}. The gray vertex ($s^x_1$) implies that, for some fixed variable $x=x' \in X$, the vertex is not included in an FD-code but is included in the FTD-code described in Lemma \ref{['aplem_NP_1']}.
• Figure 5: The set of black vertices represents an FTD-code of a path $P_n$ with $n \ge 4$ and a cycle $C_n$ with $n \ge 5$ (by joining the vertices $v_1$ and $v_n$ by an edge in each case).

Theorems & Definitions (45)

• Theorem 1
• proof
• Corollary 1
• Corollary 2
• Corollary 3
• Example 1
• Lemma 1
• proof
• Corollary 4
• Remark 1