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

TL;DR

This paper introduces visibility polynomials, explores dual mutual-visibility sets, and characterizes total mutual-visibility sets in graphs, revealing new properties and differences among various types of mutual-visibility sets.

Contribution

It defines and analyzes visibility polynomials, distinguishes dual mutual-visibility sets from others, and characterizes total mutual-visibility sets in graphs.

Findings

Every polynomial with nonnegative integer coefficients and constant term 1 is a dual visibility polynomial of some graph.

Dual mutual-visibility sets can have sizes corresponding to any subset of positive integers plus zero.

Characterizations are provided for total mutual-visibility sets and related graph properties.

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.