Exploring Algorithmic Solutions for the Independent Roman Domination Problem in Graphs

TL;DR

This work addresses the MIN-IRD problem, which seeks the Independent Roman Domination Number $i_R(G)$—the minimum total weight of an Independent Roman Dominating Function on a graph $(V,E)$ where $V_0,V_1,V_2$ partition $V$ and $i_R(G)$ is the minimum $w(f)$ over IRDFs. It develops polynomial-time (in several cases linear-time) algorithms for three graph families: distance-hereditary graphs, split graphs, and $P_4$-sparse graphs, leveraging decomposition trees, DP recurrences, and structural properties. Key results include a linear-time DP for distance-hereditary graphs computing $i_R(G)=\min\{u^0(G),u^1(G),u^2(G)\}$, a linear-time formula $i_R(G)=i(G)+1$ for split graphs, and a case-based, polynomial-time framework for $P_4$-sparse graphs using spider decompositions and join/union rules. Together these findings extend tractability boundaries for MIN-IRD on structurally rich graph classes and motivate further study on chordal and related graph families.

Abstract

Given a graph $G=(V,E)$, a function $f:V\to \{0,1,2\}$ is said to be a \emph{Roman Dominating function} if for every $v\in V$ with $f(v)=0$, there exists a vertex $u\in N(v)$ such that $f(u)=2$. A Roman Dominating function $f$ is said to be an \emph{Independent Roman Dominating function} (or IRDF), if $V_1\cup V_2$ forms an independent set, where $V_i=\{v\in V~\vert~f(v)=i\}$, for $i\in \{0,1,2\}$. The total weight of $f$ is equal to $\sum_{v\in V} f(v)$, and is denoted as $w(f)$. The \emph{Independent Roman Domination Number} of $G$, denoted by $i_R(G)$, is defined as min$\{w(f)~\vert~f$ is an IRDF of $G\}$. For a given graph $G$, the problem of computing $i_R(G)$ is defined as the \emph{Minimum Independent Roman Domination problem}. The problem is already known to be NP-hard for bipartite graphs. In this paper, we further study the algorithmic complexity of the problem. In this paper, we propose a polynomial-time algorithm to solve the Minimum Independent Roman Domination problem for distance-hereditary graphs, split graphs, and $P_4$-sparse graphs.

Figures (2)

• Figure 1: Complexity status of the MIN-IRD problem on some well known graph classes
• Figure 2: An example of a distance-hereditary graph with its decomposition tree

Theorems & Definitions (19)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Theorem 2.1
• Theorem 3.1
• Theorem 3.2
• proof