20.500.12556/RUL-174515 Visibility polynomials, dual visibility spectrum, and characterization of total mutual-visibility sets 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. mutual-visibility sets variety of mutual-visibility sets convex subgraphs integer polynomial množice vzajemne vidnosti raznolikost množic vzajemne vidnosti konveksni podgrafi celoštevilski polinom true false true Angleški jezik Slovenski jezik Članek v reviji 2025-10-03 10:23:48 2025-10-03 10:23:52 2026-06-10 16:16:09 0000-00-00 00:00:00 2025 0 0 Str. 1883–1901 iss. 4 Vol. 99 Aug. 2025 0000-00-00 Zaloznikova Objavljeno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 519.17 0001-9054 10.1007/s00010-025-01197-y 251531267 s00010-025-01197-y.pdf s00010-025-01197-y.pdf 1 1946EA5E94FA5B0654C901725CE4118A 97a970c294891fffd6ad33d7ebedaa50325d9d482c6636a81bb4cbcf1b312ea1 468d06e2-a032-11f0-9328-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=219339 https://link.springer.com/article/10.1007/s00010-025-01197-y 1 165803d5-a032-11f0-9328-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=219338 Fakulteta za matematiko in fiziko 0 0 0