The Signed Roman Domination Number of Ladder graphs, circular Ladder graphs and their complements
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| Format: | Preprint |
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2024
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| _version_ | 1866909249475444736 |
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| author | Haje, Dilbak Ahmed, Delbrin Izanloo, Hassan Saikia, Manjil |
| author_facet | Haje, Dilbak Ahmed, Delbrin Izanloo, Hassan Saikia, Manjil |
| contents | Let $G=(V,E)$ be a finite connected simple graph with vertex set $V$ and edge set $E$. A signed Roman dominating function (SRDF) on a graph $G$ is a function $f: V \rightarrow \{-1, 1, 2\}$ that satisfies two conditions: (i) $\sum_{y\in N[x]} f(y)\geq1$ for each $x\in V$, where the set $N[x]$ is the closed neighborhood of $x$ consisting of $x$ and vertices of $V$ that are adjacent to $x$, and (ii) each vertex $x\in V$ where $f(x) = -1$ is adjacent to at least one vertex $y\in V$ where $f(y)=2$. The weight of a SRDF is the sum of its function values over all vertices. The signed Roman domination number of $G$, denoted by $γ_{SR}(G)$, is the minimum weight of a SRDF on $G$. In this paper, we investigate the signed Roman domination number of the Ladder graph $LG_n$, the circular Ladder graph $CL_n$ and their complements. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_07182 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Signed Roman Domination Number of Ladder graphs, circular Ladder graphs and their complements Haje, Dilbak Ahmed, Delbrin Izanloo, Hassan Saikia, Manjil Combinatorics Let $G=(V,E)$ be a finite connected simple graph with vertex set $V$ and edge set $E$. A signed Roman dominating function (SRDF) on a graph $G$ is a function $f: V \rightarrow \{-1, 1, 2\}$ that satisfies two conditions: (i) $\sum_{y\in N[x]} f(y)\geq1$ for each $x\in V$, where the set $N[x]$ is the closed neighborhood of $x$ consisting of $x$ and vertices of $V$ that are adjacent to $x$, and (ii) each vertex $x\in V$ where $f(x) = -1$ is adjacent to at least one vertex $y\in V$ where $f(y)=2$. The weight of a SRDF is the sum of its function values over all vertices. The signed Roman domination number of $G$, denoted by $γ_{SR}(G)$, is the minimum weight of a SRDF on $G$. In this paper, we investigate the signed Roman domination number of the Ladder graph $LG_n$, the circular Ladder graph $CL_n$ and their complements. |
| title | The Signed Roman Domination Number of Ladder graphs, circular Ladder graphs and their complements |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2407.07182 |