On the Visibility Polynomial of Graphs

[2507.01851] On the Visibility Polynomial of Graphs Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Combinatorics arXiv:2507.01851 (math) [Submitted on 2 Jul 2025 ( v1 ), last revised 11 Jul 2025 (this version, v2)] Title: On the Visibility Polynomial of Graphs Authors: Tonny K B , Shikhi M View a PDF of the paper titled On the Visibility Polynomial of Graphs, by Tonny K B and 1 other authors View PDF HTML (experimental) Abstract: 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. Comments: 12 pages, 3 figures, 1 table, 1 algorithm Subjects: Combinatorics (math.CO) MSC classes: 05C31, 05C39 Cite as: arXiv:2507.01851 [math.CO] (or arXiv:2507.01851v2 [math.CO] for this version) https://doi.org/10.48550/arXiv.2507.01851 Focus to learn more arXiv-issued DOI via DataCite Journal reference: IAENG International Journal of Applied Mathematics 56 (2026), Issue 4, 1265-1274 Submission history From: Tonny K B [ view email ] [v1] Wed, 2 Jul 2025 16:06:04 UTC (15 KB) [v2] Fri, 11 Jul 2025 09:28:57 UTC (16 KB) Full-text links: Access Paper: View a PDF of the paper titled On the Visibility Polynomial of Graphs, by Tonny K B and 1 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.CO < prev | next > new | recent | 2025-07 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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