Exploring Algorithmic Solutions for the Independent Roman Domination Problem in Graphs
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| Format: | Preprint |
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2024
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| _version_ | 1866913427564265472 |
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| author | Paul, Kaustav Sharma, Ankit Pandey, Arti |
| author_facet | Paul, Kaustav Sharma, Ankit Pandey, Arti |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_03831 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Exploring Algorithmic Solutions for the Independent Roman Domination Problem in Graphs Paul, Kaustav Sharma, Ankit Pandey, Arti Combinatorics Discrete Mathematics 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. |
| title | Exploring Algorithmic Solutions for the Independent Roman Domination Problem in Graphs |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2407.03831 |