Algorithmic Results for Weak Roman Domination Problem in Graphs

TL;DR

The paper investigates algorithmic aspects of the Minimum Weak Roman Domination (MIN-WRD) problem and its decision variant (DECIDE-WRD). It provides a polynomial-time $O(n^3)$-time algorithm for MIN-WRD on $P_4$-sparse graphs by exploiting join and spider decompositions, and establishes NP-hardness for DECIDE-WRD in comb convex and star convex bipartite graphs as well as for split graphs. It also delivers approximation results, showing a $2(1+\ln(\Delta+1))$-factor algorithm and proving APX-hardness for degree-4 graphs. Collectively, the work maps the boundary between tractable and intractable instances across several graph families and offers practical approximation strategies.

Abstract

Consider a graph $G = (V, E)$ and a function $f: V \rightarrow \{0, 1, 2\}$. A vertex $u$ with $f(u)=0$ is defined as \emph{undefended} by $f$ if it lacks adjacency to any vertex with a positive $f$-value. The function $f$ is said to be a \emph{Weak Roman Dominating function} (WRD function) if, for every vertex $u$ with $f(u) = 0$, there exists a neighbour $v$ of $u$ with $f(v) > 0$ and a new function $f': V \rightarrow \{0, 1, 2\}$ defined in the following way: $f'(u) = 1$, $f'(v) = f(v) - 1$, and $f'(w) = f(w)$, for all vertices $w$ in $V\setminus\{u,v\}$; so that no vertices are undefended by $f'$. The total weight of $f$ is equal to $\sum_{v\in V} f(v)$, and is denoted as $w(f)$. The \emph{Weak Roman Domination Number} denoted by $γ_r(G)$, represents $min\{w(f)~\vert~f$ is a WRD function of $G\}$. For a given graph $G$, the problem of finding a WRD function of weight $γ_r(G)$ is defined as the \emph{Minimum Weak Roman domination problem}. The problem is already known to be NP-hard for bipartite and chordal graphs. In this paper, we further study the algorithmic complexity of the problem. We prove the NP-hardness of the problem for star convex bipartite graphs and comb convex bipartite graphs, which are subclasses of bipartite graphs. In addition, we show that for the bounded degree star convex bipartite graphs, the problem is efficiently solvable. We also prove the NP-hardness of the problem for split graphs, a subclass of chordal graphs. On the positive side, we give polynomial-time algorithms to solve the problem for $P_4$-sparse graphs. Further, we have presented some approximation results.

Figures (4)

• Figure 1: Example of a WRD function on $C_4$
• Figure 2: The Construction $\mathcal{A}_1$ applied on the Example 1
• Figure 3: The Construction $\mathcal{A}_2$ applied on the Example 1
• Figure 4: The Construction $\mathcal{A}_3$ applied on the Example 1

Theorems & Definitions (33)

• Theorem 2.1
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Theorem 2.2