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

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

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.

• Jezik: Angleški jezik
• Ključne besede: mutual-visibility sets, variety of mutual-visibility sets, convex subgraphs, integer polynomial
• Vrsta gradiva: Članek v reviji
• Tipologija: 1.01 - Izvirni znanstveni članek
• Organizacija: FMF - Fakulteta za matematiko in fiziko
• Status publikacije: Objavljeno
• Različica publikacije: Objavljena publikacija
• Datum objave: 01.08.2025
• Leto izida: 2025
• Št. strani: Str. 1883–1901
• Številčenje: Vol. 99, iss. 4
• PID: 20.500.12556/RUL-174515
• UDK: 519.17
• ISSN pri članku: 0001-9054
• DOI: 10.1007/s00010-025-01197-y
• COBISS.SI-ID: 251531267
• Datum objave v RUL: 03.10.2025

Gradivo je del revije

• Naslov: Aequationes mathematicae
• Skrajšan naslov: Aequ. math.
• Založnik: Springer
• ISSN: 0001-9054
• COBISS.SI-ID: 1327364

Sekundarni jezik

• Jezik: Slovenski jezik
• Ključne besede: množice vzajemne vidnosti, raznolikost množic vzajemne vidnosti, konveksni podgrafi, celoštevilski polinom