Visibility polynomial of corona of two graphs
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866916930193981440 |
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| author | B, Tonny K M, Shikhi |
| author_facet | B, Tonny K M, Shikhi |
| contents | In multiagent systems, effective coordination, coverage, and communication often rely on the concept of visibility between agents or nodes within the system. Graph-theoretically, for any subset $X$ of vertices of a graph $G$, two vertices are said to be $X$-visible if there exists a shortest path between them that contains no vertex of $X$ as an internal vertex. In this paper, we investigate the visibility polynomial associated with the corona product of two graphs. The visibility polynomial encodes the number of mutual-visibility sets of all orders within a graph, and the process of enumerating these sets provides a deeper understanding of their structural properties. We characterize the structure of mutual-visibility sets arising specifically within the corona product. As part of this study, we introduce the notion of $C_Q$-visible sets, defined with respect to a selected subset $Q$ of vertices in a graph $G$. A $C_Q$-visible set is a collection of vertices in $\overline{Q}$ that is not only $Q$-visible, but also individually visible from each vertex in $Q$. Using this concept, we establish several characterizations and properties of mutual-visibility sets within the corona product, thereby providing deeper insights into their structure and behavior. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_02509 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Visibility polynomial of corona of two graphs B, Tonny K M, Shikhi Combinatorics 05C30, 05C31, 05C76 In multiagent systems, effective coordination, coverage, and communication often rely on the concept of visibility between agents or nodes within the system. Graph-theoretically, for any subset $X$ of vertices of a graph $G$, two vertices are said to be $X$-visible if there exists a shortest path between them that contains no vertex of $X$ as an internal vertex. In this paper, we investigate the visibility polynomial associated with the corona product of two graphs. The visibility polynomial encodes the number of mutual-visibility sets of all orders within a graph, and the process of enumerating these sets provides a deeper understanding of their structural properties. We characterize the structure of mutual-visibility sets arising specifically within the corona product. As part of this study, we introduce the notion of $C_Q$-visible sets, defined with respect to a selected subset $Q$ of vertices in a graph $G$. A $C_Q$-visible set is a collection of vertices in $\overline{Q}$ that is not only $Q$-visible, but also individually visible from each vertex in $Q$. Using this concept, we establish several characterizations and properties of mutual-visibility sets within the corona product, thereby providing deeper insights into their structure and behavior. |
| title | Visibility polynomial of corona of two graphs |
| topic | Combinatorics 05C30, 05C31, 05C76 |
| url | https://arxiv.org/abs/2509.02509 |