On the Visibility Polynomial of Graphs
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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_ | 1866918436047683584 |
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| author | B, Tonny K M, Shikhi |
| author_facet | B, Tonny K M, Shikhi |
| contents | Let G(V,E) be a simple graph and let X subset of V. Two vertices u and v are said to be X-visible if there exists a shortest u,v-path P such that V(P) intersection X is a subset of {u, v}. A set X is called a mutual-visibility set of G if every pair of vertices in X are X-visible. The visibility polynomial of a graph G is defined as nu (G)=sum_{i >= 0} r_i x^i, where r_i denotes the number of mutual-visibility sets in G of cardinality i. In the present paper, the visibility polynomial is studied for some well-known classes of graphs. In particular, the instance at which the number of maximal mutual-visibility sets is equal for cycle graphs is identified. The visibility polynomial of the join of two graphs is studied. The algorithm for computing the visibility polynomial of a graph has been identified to have a time complexity of O(n^3.2^n) making the problem computationally intensive for larger graphs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_01851 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Visibility Polynomial of Graphs B, Tonny K M, Shikhi Combinatorics 05C31, 05C39 Let G(V,E) be a simple graph and let X subset of V. Two vertices u and v are said to be X-visible if there exists a shortest u,v-path P such that V(P) intersection X is a subset of {u, v}. A set X is called a mutual-visibility set of G if every pair of vertices in X are X-visible. The visibility polynomial of a graph G is defined as nu (G)=sum_{i >= 0} r_i x^i, where r_i denotes the number of mutual-visibility sets in G of cardinality i. In the present paper, the visibility polynomial is studied for some well-known classes of graphs. In particular, the instance at which the number of maximal mutual-visibility sets is equal for cycle graphs is identified. The visibility polynomial of the join of two graphs is studied. The algorithm for computing the visibility polynomial of a graph has been identified to have a time complexity of O(n^3.2^n) making the problem computationally intensive for larger graphs. |
| title | On the Visibility Polynomial of Graphs |
| topic | Combinatorics 05C31, 05C39 |
| url | https://arxiv.org/abs/2507.01851 |