General position polynomials

TL;DR

This paper introduces the general position polynomial $\psi(G)=\sum_{i\ge0} a_i x^i$, where $a_i$ counts general position sets of size $i$ in a graph $G$, and studies its behavior across graph families and operations. It establishes base polynomials for standard graphs, presents an inclusion-exclusion formula to compute $\psi(G)$ from maximal GP-sets, and analyzes how $\psi$ transforms under disjoint unions, joins, and Cartesian products, including explicit grid formulas with degree up to 4. The authors show that unimodality of $\psi(G)$ fails in general and even on trees, but holds for several significant classes such as combs, Kneser graphs $K(n,2)$, and certain multipartite graphs, while providing substantial partial results for other families. The work highlights rich combinatorial structure, connects to clique polynomials, and raises open problems about extensions and related position notions, offering a foundation for further study of GP-sets via generating polynomials.

Abstract

A subset of vertices of a graph $G$ is a general position set if no triple of vertices from the set lie on a common shortest path in $G$. In this paper we introduce the general position polynomial as $\sum_{i \geq 0} a_i x^i$, where $a_i$ is the number of distinct general position sets of $G$ with cardinality $i$. The polynomial is considered for several well-known classes of graphs and graph operations. It is shown that the polynomial is not unimodal in general, not even on trees. On the other hand, several classes of graphs, including Kneser graphs $K(n,2)$, with unimodal general position polynomials are presented.

Figures (2)

• Figure 1: Trees $T_1^{(1)}$ and $T_2^{(1)}$.
• Figure 2: The broom $B_{4,6}$.

Theorems & Definitions (18)

• Definition 1.1
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Proposition 3.1
• Proposition 3.2
• proof
• Proposition 3.3
• Corollary 3.4