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arXiv · 2412.03066

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

Csilla Bujtás·Sandi Klavžar·Jing Tian
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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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Csilla Bujtás, Sandi Klavžar, Jing Tian. 2024-12-04. Visibility polynomials, dual visibility spectrum, and characterization of total mutual-visibility sets. https://doi.org/10.1007/s00010-025-01197-y

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