Minimally unbalanced diamond-free graphs and Dyck-paths

TL;DR

This paper characterizes minimally unbalanced diamond-free graphs and reveals a connection between these graphs and Dyck-paths, advancing understanding of balanced graph structures.

Contribution

It provides the first characterization of minimally unbalanced diamond-free graphs and links them to Dyck-paths, a novel structural insight.

Findings

Characterization of minimally unbalanced diamond-free graphs

Establishment of a connection with Dyck-paths

Advancement in understanding balanced graph structures

Abstract

A $\{0,1\}$-matrix $\mathsf{A}$ is balanced if it does not contain a submatrix of odd order having exactly two 1's per row and per column. A graph is balanced if its clique-matrix is balanced. No characterization of minimally unbalanced graphs is known, and even no conjecture on the structure of such graphs has been posed, contrarily to what happened for perfect graphs. In this paper, we provide such a characterization for the class of diamond-free graphs and establish a connection between minimally unbalanced diamond-free graphs and Dyck-paths.