Inversion diameter and 2-edge-colored homomorphisms

2026-02-27

AI Summary

This study investigates the maximum transformation distance—referred to as the reversal diameter—of graphs under vertex subset reversal operations, establishing upper and lower bounds across various graph classes. The work innovatively models the reversal operation as a homomorphism problem on 2-edge-colored graphs and introduces, for the first time, the notion of homomorphism-universal 2-edge-colored graphs. This approach improves the known upper bound on the reversal diameter from quadratic to linear in terms of the acyclic chromatic number. Key contributions include proving a lower bound of 6 for planar graphs, resolving the case of graphs with girth 7, deriving an upper bound of Δ + log Δ for triangle-free graphs, and establishing a general upper bound for subdivision graphs.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in 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: Measurement, analysis, and circumvention of Web censorship

Abstract

In an oriented graph, the inversion of a subset of vertices X is the operation reversing the direction of every arc with both endpoints in X. Given a graph G, the inversion distance between two orientations G is the minimum number of inversions transforming one into the other. The inversion diameter diam(G) is the maximum such distance over all pairs of orientations of G. Through an equivalent formulation of inversions over 2-edge-colorings of G, we introduce the use of homomorphism-universal 2-edge-colored graphs to obtain bounds on the inversion diameter of various classes of graphs. Our first result upper bounds the inversion diameter by a linear function of the acyclic chromatic number, improving on the previous quadratic dependency. We then consider the inversion diameter of planar graphs, exhibiting a lower bound of 6, as well as new lower and upper bounds for those of a given girth, in particular settling the girth 7 case. We then show that any triangle-free graph G with maximum degree D satisfies diam(G)<= D + log D, making progress on the conjecture of Havet et al. that diam(G)<= D. Finally, we prove a general result about subdivisions: if a graph has inversion diameter k, any of its subdivisions has inversion diameter at most k + log k + 5.

Problem

• inversion diameter
• oriented graph
• 2-edge-colored homomorphism
• acyclic chromatic number
• graph subdivision

Innovation

• inversion diameter
• 2-edge-colored homomorphisms
• acyclic chromatic number
• planar graphs
• graph subdivisions