| Title: | Visibility polynomials, dual visibility spectrum, and characterization of total mutual-visibility sets |
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| Authors: | ID Bujtás, Csilla (Author)
ID Klavžar, Sandi (Author)
ID Tian, Jing (Author) |
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| Files: |  PDF - Presentation file, download (424,32 KB)
MD5: 1B493CB1CF4F1C1A732EBC0B7129170D
 URL - Source URL, visit https://link.springer.com/article/10.1007/s00010-025-01197-y
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| Language: | English |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: |  IMFM - Institute of Mathematics, Physics, and Mechanics
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| Abstract: | Mutual-visibility sets were motivated by visibility in distributed systems and social networks, and intertwine with several classical mathematical areas. Monotone properties of the variety of mutual-visibility sets, and restrictions of such sets to convex and isometric subgraphs are studied. Dual mutual-visibility sets are shown to be intrinsically different from other types of mutual-visibility sets. It is proved that for every finite subset $Z$ of positive integers there exists a graph $G$ that has a dual mutual-visibility set of size ▫$i$▫ if and only if $i\in Z\cup \{0\}$, while for the other types of mutual-visibility such a set consists of consecutive integers. Visibility polynomials are introduced and their properties derived. As a surprise, every polynomial with nonnegative integer coefficients and with a constant term $1$ is a dual visibility polynomial of some graph. Characterizations are given for total mutual-visibility sets, for graphs with total mutual-visibility number $1$, and for sets which are not total mutual-visibility sets, yet every proper subset is such. Along the way an earlier result from the literature is corrected. |
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| Keywords: | mutual-visibility sets, variety of mutual-visibility sets, convex subgraphs, integer polynomial |
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| Publication status: | Published |
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| Publication version: | Version of Record |
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| Publication date: | 01.08.2025 |
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| Year of publishing: | 2025 |
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| Number of pages: | str. 1883–1901 |
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| Numbering: | Vol. 99, iss. 4 |
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| PID: | 20.500.12556/DiRROS-23804  |
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| UDC: | 519.17 |
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| ISSN on article: | 0001-9054 |
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| DOI: | 10.1007/s00010-025-01197-y |
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| COBISS.SI-ID: | 251531267 |
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| Note: |
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| Publication date in DiRROS: | 03.10.2025 |
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| Views: | 252 |
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| Downloads: | 128 |
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