Revisiting the outer-weakly convex domination number in graph products
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866908990344003584 |
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| author | Anand, Bijo S. V., Ullas Chandran S. Dayap, Jonecis A. Casinillo, Leomarich F. Yap, Karen Luz P. |
| author_facet | Anand, Bijo S. V., Ullas Chandran S. Dayap, Jonecis A. Casinillo, Leomarich F. Yap, Karen Luz P. |
| contents | Let $G = (V, E)$ be a simple undirected connected graph. A set $C \subseteq V(G)$ is weakly convex in $G$ if for every two vertices $u,v$ in $G$, there exists a $u-v$ geodesic whose vertices are in $C$. A set $C \subseteq V$ is an outer-weakly convex dominating set if every vertex not in $C$ is adjacent to some vertex in $C$ and the set $V(G)\setminus C$ is weakly convex in $G$. The outer-weakly convex domination number of graph $G$, denoted by $\widetilde{ γ}_{wcon}(G)$, is the minimum cardinality of an outer-weakly convex dominating set of graph $G$. In this paper, we determine the outer-weakly convex domination number of two graphs under the Cartesian, strong and lexicographic products, and discuss some important combinatorial findings. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_15524 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Revisiting the outer-weakly convex domination number in graph products Anand, Bijo S. V., Ullas Chandran S. Dayap, Jonecis A. Casinillo, Leomarich F. Yap, Karen Luz P. Combinatorics 05C38, 05C76, 05C99, 52A01 Let $G = (V, E)$ be a simple undirected connected graph. A set $C \subseteq V(G)$ is weakly convex in $G$ if for every two vertices $u,v$ in $G$, there exists a $u-v$ geodesic whose vertices are in $C$. A set $C \subseteq V$ is an outer-weakly convex dominating set if every vertex not in $C$ is adjacent to some vertex in $C$ and the set $V(G)\setminus C$ is weakly convex in $G$. The outer-weakly convex domination number of graph $G$, denoted by $\widetilde{ γ}_{wcon}(G)$, is the minimum cardinality of an outer-weakly convex dominating set of graph $G$. In this paper, we determine the outer-weakly convex domination number of two graphs under the Cartesian, strong and lexicographic products, and discuss some important combinatorial findings. |
| title | Revisiting the outer-weakly convex domination number in graph products |
| topic | Combinatorics 05C38, 05C76, 05C99, 52A01 |
| url | https://arxiv.org/abs/2501.15524 |