On the complexity of global Roman domination problem in graphs

TL;DR

This work investigates the global Roman domination problem (GRD) and its relation to Roman domination (RD), showing that RD and GRD are not computationally equivalent by constructing two graph classes with opposite complexity profiles. It proves GRD is NP-complete on split graphs and several other restricted classes, while RD is linear-time solvable on a complementary class, and it establishes a linear-time GRD algorithm for cographs using cotree decompositions. The results map out a nuanced complexity landscape for GRD, identifying both hard and tractable graph families and prompting open questions on other graph classes and parameterized complexity. The findings have implications for understanding domination-type problems in graph theory and for practical algorithm design on structured graphs.

Abstract

A Roman dominating function of a graph $G=(V,E)$ is a labeling $f: V \rightarrow{} \{0 ,1, 2\}$ such that for each vertex $u \in V$ with $f(u) = 0$, there exists a vertex $v \in N(u)$ with $f(v) =2$. A Roman dominating function $f$ is a global Roman dominating function if it is a Roman dominating function for both $G$ and its complement $\overline{G}$. The weight of $f$ is the sum of $f(u)$ over all the vertices $u \in V$. The objective of Global Roman Domination problem is to find a global Roman dominating function with minimum weight. The objective of Global Roman Domination is to compute a global Roman dominating function of minimum weight. In this paper, we study the algorithmic aspects of Global Roman Domination problem on various graph classes and obtain the following results. 1. We prove that Roman domination and Global Roman Domination problems are not computationally equivalent by identifying graph classes on which one is linear-time solvable, while the other is NP-complete. 2. We show that Global Roman Domination problem is NP-complete on split graphs, thereby resolving an open question posed by Panda and Goyal [Discrete Applied Mathematics, 2023]. 3. We prove that Global Roman Domination problem is NP-complete on chordal bipartite graphs, planar bipartite graphs with maximum degree five and circle graphs. 4. On the positive side, we present a linear-time algorithm for Global Roman domination problem on cographs.

Figures (3)

• Figure 1: (left) A $3$-regular graph $G$; (right) The construction of $G' \in \mathcal{F}$ from $G$. Only the adjacency of a vertex $u_1 \in G_1$ to its neighbors in $G_2$ and $G_3$ is shown in $G'$.
• Figure 2: Graph $G' \in \mathcal{G}$ constructed from an X4C problem instance: $X = \{1,2,3,4,5,6, 7, 8\}$ and $C$ = $\{C_1 = \{1,2,3,4\}$, $C_2 = \{2,4,6,7\}$, $C_3 = \{3,5,6,7\}$ and $C_4 = \{5,6,7,8\}\}$. A thick edge between a vertex $v$ and a set $Y$ indicates that $v$ is adjacent to every vertex of set $Y$. The edges between the vertices of set $A$ are not shown in the figure. All the vertices of the set $S$ are not shown in the figure due to space constraints.
• Figure 3: Reduced instance of GRD problem constructed from an X3C instance: $X = \{1,2,3,4,5,6\}$ and $C$ = $\{C_1 = \{1,2,4\}$, $C_2 = \{2,3,4\}$, $C_3 = \{3,5,6\}$ and $C_4 = \{4,5,6\}\}$. The vertices of $A$ form a clique, which is not shown in the figure.

Theorems & Definitions (53)

• Lemma 1
• proof
• Lemma 2
• proof
• Theorem 3.1
• Theorem 3.2: Kikuno
• Lemma 3
• proof
• Lemma 4
• proof