Visibility polynomials, dual visibility spectrum, and characterization of total mutual-visibility sets

Bujtás, Csilla (author), Klavžar, Sandi (author), Tian, Jing (author)

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.

• Language: English
• Keywords: mutual-visibility sets, variety of mutual-visibility sets, convex subgraphs, integer polynomial
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Publication date: 01.08.2025
• Year: 2025
• Number of pages: Str. 1883–1901
• Numbering: Vol. 99, iss. 4
• PID: 20.500.12556/RUL-174515
• UDC: 519.17
• ISSN on article: 0001-9054
• DOI: 10.1007/s00010-025-01197-y
• COBISS.SI-ID: 251531267
• Publication date in RUL: 03.10.2025

Record is a part of a journal

• Title: Aequationes mathematicae
• Shortened title: Aequ. math.
• Publisher: Springer
• ISSN: 0001-9054
• COBISS.SI-ID: 1327364

Secondary language

• Language: Slovenian
• Keywords: množice vzajemne vidnosti, raznolikost množic vzajemne vidnosti, konveksni podgrafi, celoštevilski polinom