On the Visibility Polynomial of Graphs

TL;DR

This paper introduces the concept of the visibility polynomial in graphs, analyzes it for specific graph classes, and discusses computational complexity for calculating it.

Contribution

It defines the visibility polynomial, studies its properties for well-known graphs, and examines the complexity of computing it.

Findings

Identified when the number of maximal mutual-visibility sets is equal in cycle graphs.

Derived the visibility polynomial for the join of two graphs.

Presented an algorithm with O(n^3.2^n) complexity for computing the visibility polynomial.

Abstract

Let G(V,E) be a simple graph and let X subset of V. Two vertices u and v are said to be X-visible if there exists a shortest u,v-path P such that V(P) intersection X is a subset of {u, v}. A set X is called a mutual-visibility set of G if every pair of vertices in X are X-visible. The visibility polynomial of a graph G is defined as nu (G)=sum_{i >= 0} r_i x^i, where r_i denotes the number of mutual-visibility sets in G of cardinality i. In the present paper, the visibility polynomial is studied for some well-known classes of graphs. In particular, the instance at which the number of maximal mutual-visibility sets is equal for cycle graphs is identified. The visibility polynomial of the join of two graphs is studied. The algorithm for computing the visibility polynomial of a graph has been identified to have a time complexity of O(n^3.2^n) making the problem computationally intensive for larger graphs.