2024/12/04 by Csilla Bujtás, Bujtás, Csilla, Sandi Klavžar +3 · 6 citations
Computer Science · #Combinatorics (math.CO) #FOS: Mathematics #Image Enhancement Techniques #Video Surveillance and Tracking Methods
paper · pdf · doi:10.48550/arxiv.2412.03066
openalex publication_date 2024/12/04 · openalex created_date 2024/12/06 · openalex updated_date 2026/07/28
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∈ Z∪ \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.