The Interplay Between Domination and Separation in Graphs

TL;DR

This paper addresses the complexity of finding minimum separating sets in a graph and study the interplay of these separation properties with several codes as well as the interplay of separation and complementation.

Abstract

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).

Figures (8)

• Figure 1: Minimum S-sets and X-codes in a graph (the black vertices belong to the S-sets and X-codes), where (a) is both a D-set and an L-set and thus an LD-code, (b) is both a TD-set and an I-set, hence also both an ID- and an ITD-code, (c) is an O-set and thus an OD-code, (d) is an F-set, hence both an FD- and an FTD-code as well as an ITD-code.
• Figure 2: The relations between the X-numbers for all $X \in {\textsc{Codes}}$, where $X' \longrightarrow X$ stands for $\gamma^{X'}(G) \leq \gamma^X(G)$.
• Figure 3: Minimum D-sets and TD-sets in two graphs having a universal vertex (the black vertices are universal and form the D-sets, the grey vertices complete the D-sets to the TD-sets), where (a) is the clique $K_4$, (b) is the star $K_{1,3}$.
• Figure 4: Illustration of the graph $G_I$ constructed by Reduction \ref{['red:general']}. Here $U = \{u_1, u_2, \ldots u_z\}$ and $\mathcal{T}\xspace = \{T_1, T_2, \ldots , T_y\}$. The set of black vertices in Figure \ref{['fig:NP_C_all']} is an example of an I-set of $G_I$.
• Figure 5: Minimum S-sets in the thin headless spider $H_4$ (the black vertices belong to the S-sets), where (a) is both an L-set and an O-set, (b) is an I-set, (c) is an F-set.

Theorems & Definitions (29)

• Lemma 2.1
• Theorem 2.2
• proof : Proof (sketch)
• Theorem 2.3
• Theorem 3.1
• Theorem 3.2
• Theorem 3.3
• Theorem 3.4
• Theorem 4.1
• Corollary 4.2