Improved Total Domination and Total Roman Domination in Unit Disk Graphs

TL;DR

This work investigates total domination and total Roman domination in geometric unit disk graphs (UDGs). It establishes NP-completeness of the total Roman domination set (TRDS) problem in UDGs, using a reduction from Dominating Set on grid graphs, and then presents two geometry-aware approximation schemes: a 7.17-factor algorithm for the total dominating set (TDS) and a 6.03-factor algorithm for TRDS, both with running time $O(n\log k)$ where $k$ bounds the relevant independent set size. The methods leverage maximal independent sets in the geometric setting and reduce the augmentation step to greedy set cover, guided by packing lemmas (e.g., at most 5 neighbors in certain configurations). These results advance the understanding of domination problems in UDGs by offering near-constant-factor guarantees with efficient runtimes, complementing prior work in the area.

Abstract

Let $G=(V, E)$ be a simple undirected graph with no isolated vertex. A set $D_t\subseteq V$ is a total dominating set of $G$ if $(i)$ $D_t$ is a dominating set, and $(ii)$ the set $D_t$ induces a subgraph with no isolated vertex. The total dominating set of minimum cardinality is called the minimum total dominating set, and the size of the minimum total dominating set is called the total domination number ($γ_t(G)$). Given a graph $G$, the total dominating set (TDS) problem is to find a total dominating set of minimum cardinality. A Roman dominating function (RDF) on a graph $G$ is a function $f:V\rightarrow \{0,1,2\}$ such that each vertex $v\in V$ with $f(v)=0$ is adjacent to at least one vertex $u\in V$ with $f(u)=2$. A RDF $f$ of a graph $G$ is said to be a total Roman dominating function (TRDF) if the induced subgraph of $V_1\cup V_2$ does not contain any isolated vertex, where $V_i=\{u\in V|f(u)=i\}$. Given a graph $G$, the total Roman dominating set (TRDS) problem is to minimize the weight, $W(f)=\sum_{u\in V} f(u)$, called the total Roman domination number ($γ_{tR}(G)$). In this paper, we are the first to show that the TRDS problem is NP-complete in unit disk graphs (UDGs). Furthermore, we propose a $7.17\operatorname{-}$ factor approximation algorithm for the TDS problem and a $6.03\operatorname{-}$ factor approximation algorithm for the TRDS problem in geometric unit disk graphs. The running time for both algorithms is notably bounded by $O(n\log{k})$, where $n$ represents the number of vertices in the given UDG and $k$ represents the size of the independent set in (i.e., $D$ and $V_2$ in TDS and TRDS problems, respectively) the given UDG.

Figures (2)

• Figure 1: Illustration of $G$ and $G'$
• Figure 2: Illustration of $D$ and $V\setminus D$

Theorems & Definitions (24)

• Lemma 1
• Lemma 2
• Theorem 1
• Theorem 2
• Lemma 3
• proof
• Lemma 4
• proof
• Theorem 3
• proof