On the Complexity of Signed Roman Domination

TL;DR

The paper investigates the computational complexity of Signed Roman Domination (SRD), establishing a spectrum of results across classical and parameterized complexity. It proves NP-completeness on split graphs, shows W[2]-hardness by weight (even on bipartite graphs), and W[1]-hardness by feedback vertex set number (and thus treewidth/clique-width), while providing FPT algorithms for neighbourhood diversity and vertex cover number. A kernelization lower bound is given: SRD does not admit a polynomial kernel parameterized by vertex cover unless coNP ⊆ NP/poly. The authors derive these results through reductions from Dominating Set and MRSS, and via ILP-based FPT techniques, highlighting a distinct parameterized landscape for SRD compared to RD and SD and outlining several open directions for future work.

Abstract

Given a graph $G = (V, E)$, a signed Roman dominating function is a function $f: V \rightarrow \{-1, 1, 2\}$ such that for every vertex $u \in V$: $\sum_{v \in N[u]} f(v) \geq 1$ and for every vertex $u \in V$ with $f(u) = -1$, there exists a vertex $v \in N(u)$ with $f(v) = 2$. The weight of a signed Roman dominating function $f$ is $\sum_{u \in V} f(u)$. The objective of \srd{} (SRD) problem is to compute a signed Roman dominating function with minimum weight. The problem is known to be NP-complete even when restricted to bipartite graphs and planar graphs. In this paper, we advance the complexity study by showing that the problem remains NP-complete on split graphs. In the realm of parameterized complexity, we prove that the problem is W[2]-hard parameterized by weight, even on bipartite graphs. We further show that the problem is W[1]-hard parameterized by feedback vertex set number (and hence also when parameterized by treewidth or clique-width). On the positive side, we present an FPT algorithm parameterized by neighbourhood diversity (and by vertex cover number). Finally, we complement this result by proving that the problem does not admit a polynomial kernel parameterized by vertex cover number unless coNP $\subseteq$ NP/poly.

Figures (7)

• Figure 1: Hasse diagram representing the relation between various structural parameters. A directed edge from a parameter $k_1$ to $k_2$ indicates that $k_1 \leq f(k_2)$ for some computable function $f$.
• Figure 2: Reduced instance $G'$ from an instance of the dominating set $G$ and $k = 2$. Edges between the vertices of the clique $A \cup B \cup C \cup D \cup E$ are not shown in the figure. The edges incident on the vertex $x_1$ are colored in blue.
• Figure 3: Construction of $G'$ from $G$. Here, $D$ is used as a placeholder for $d(v)+1$.
• Figure 4: Circle representation of $G'$
• Figure 5: Reduced instance of SRD problem constructed from MRSS problem instance $S =\{(2, 1), (1, 2), (1, 1)\}, t= (3, 3), k = 2, m = 2.$

Theorems & Definitions (44)

• Definition 1: Split graphs
• Definition 2: Circle graphs
• Definition 3: Neighbourhood diversity
• Definition 4: Vertex cover number
• Definition 5: Feedback vertex set number
• Theorem 3.1: Kikuno
• Lemma 1
• proof
• Theorem 3.2
• Corollary 1