Posets of Geometric Graphs

TL;DR

This paper introduces a partial order on geometric graph realizations using homomorphisms, characterizes their structure for small graphs, and explores properties of these posets for various classes of graphs.

Contribution

It develops tools to compare geometric realizations of graphs via homomorphisms and provides classifications and Hasse diagrams for small cases, extending understanding of geometric graph realizations.

Findings

Classified geometric realizations of P_n, C_n, K_n for 3 ≤ n ≤ 6.

Identified minimal and maximal elements in the posets of P_n and C_n.

Established subposet relations and chain lengths in the posets for various graphs.

Abstract

A geometric graph G(bar) is a simple graph drawn in the plane, on points in general position, with straight-line edges. We call G(bar) a geometric realization of the underlying abstract graph G. A geometric homomorphism is a vertex map that preserves adjacencies and crossings (but not necessarily non-adjacencies or non-crossings). This work uses geometric homomorphisms to introduce a partial order on the set of isomorphism classes of geometric realizations of an abstract graph G. We say G(bar) precedes G(hat) if G(bar) and G(hat) are geometric realizations of G and there is a vertex-injective geometric homomorphism from G(bar) to G(hat). This paper develops tools to determine when two geometric realizations are comparable. Further, for 3 \leq n \leq 6, this paper provides the isomorphism classes of geometric realizations of P_n, C_n and K_n, as well as the Hasse diagrams of the geometric homomorphism posets of these graphs. The paper also provides the following results for general n: the poset of P_n and C_n has a unique minimal element and a unique maximal element; if k \leq n then the poset of P_k (resp., the poset of C_k) is a subposet of the poset for P_n (resp., C_n); and the poset for K_n contains a chain of length n-2.