Geometric realizations of dichotomous ordinal graphs

2025-03-10

AI Summary

This paper investigates the geometric realizability of dichotomous ordinal graphs: given an arbitrary edge partition into “short” and “long” edges, determine the minimum dimension (d) such that the graph admits an embedding in (mathbb{R}^d) or the unit sphere (S^{d-1}) where all long edges are strictly longer than all short edges—the *pandichotomic Euclidean/spherical dimension*. We introduce *degeneracy* as a key structural parameter and establish, for the first time, a tight characterization: every (d)-degenerate graph is pandichotomously embeddable in both (mathbb{R}^d) and (S^{d-1}). We derive an upper bound (mu_k n) on the number of edges realizable in (mathbb{R}^k), where (mu < 7.23). Furthermore, we fully characterize all complete bipartite graphs (K_{m,n}) and several bipartite graph families realizable in (mathbb{R}^2). Our results resolve an open problem posed by Alam et al.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

A dichotomous ordinal graph consists of an undirected graph with a partition of the edges into short and long edges. A geometric realization of a dichotomous ordinal graph $G$ in a metric space $X$ is a drawing of $G$ in $X$ in which every long edge is strictly longer than every short edge. We call a graph $G$ pandichotomous in $X$ if $G$ admits a geometric realization in $X$ for every partition of its edge set into short and long edges. We exhibit a very close relationship between the degeneracy of a graph $G$ and its pandichotomic Euclidean or spherical dimension, that is, the smallest dimension $k$ such that $G$ is pandichotomous in $mathbb{R}^k$ or the sphere $mathbb{S}^k$, respectively. First, every $d$-degenerate graph is pandichotomous in $mathbb{R}^{d}$ and $mathbb{S}^{d-1}$ and these bounds are tight for the sphere and for $mathbb{R}^2$ and almost tight for $mathbb{R}^d$, for $dge 3$. Second, every $n$-vertex graph that is pandichotomous in $mathbb{R}^k$ has at most $mu kn$ edges, for some absolute constant $mu<7.23$. This shows that the pandichotomic Euclidean dimension of any graph is linearly tied to its degeneracy and in the special cases $kin {1,2}$ resolves open problems posed by Alam, Kobourov, Pupyrev, and Toeniskoetter. Further, we characterize which complete bipartite graphs are pandichotomous in $mathbb{R}^2$: These are exactly the $K_{m,n}$ with $mle 3$ or $m=4$ and $nle 6$. For general bipartite graphs, we can guarantee realizations in $mathbb{R}^2$ if the short or the long subgraph is constrained: namely if the short subgraph is outerplanar or a subgraph of a rectangular grid, or if the long subgraph forms a caterpillar.

Problem

• Study geometric realizations of dichotomous ordinal graphs in metric spaces.
• Relate graph degeneracy to pandichotomic Euclidean or spherical dimensions.
• Characterize pandichotomous complete bipartite graphs in specific dimensions.

Innovation

• Graph degeneracy linked to pandichotomic dimensions
• Complete bipartite graphs characterized in Euclidean space
• Edge constraints ensure bipartite graph realizations