Characterization and linear-time recognition of balanced distance-hereditary graphs

2026-07-01

AI Summary

This work addresses the long-standing challenge of characterizing and efficiently recognizing balanced graphs within the class of distance-hereditary graphs, which previously lacked a concise forbidden induced subgraph characterization. The paper establishes that a distance-hereditary graph is balanced if and only if it is hereditary clique-Helly, yielding a minimal characterization involving a single forbidden induced subgraph, $\overline{3K_2}$. Building on this structural theorem, the authors devise a linear-time recognition algorithm that, when the input graph is not balanced, explicitly returns an induced $\overline{3K_2}$ as a certificate of non-balancedness. This result provides the first single-forbidden-subgraph characterization of balanced graphs in the distance-hereditary setting, resolving both the structural description and efficient recognition problems for this subclass.

Abstract

A graph is balanced if its clique-matrix contains no square submatrix of odd order with exactly two $1$'s in each row and in each column. Although it is known that a graph is balanced if and only if it contains no induced extended odd sun, a characterization of balanced graphs by minimal forbidden induced subgraphs is still unknown. In this work, we prove that, within the class of distance-hereditary graphs, balanced graphs are exactly the hereditary clique-Helly graphs. Equivalently, they are characterized by a single forbidden induced subgraph, namely $\overline{3K_2}$. From this result, we derive an explicit linear-time algorithm that decides balancedness within the class of distance-hereditary graphs and returns an induced $\overline{3K_2}$ when the input graph is not balanced.

Problem

• balanced graphs
• distance-hereditary graphs
• forbidden induced subgraphs
• clique-Helly graphs

Innovation

• balanced graphs
• distance-hereditary graphs
• hereditary clique-Helly graphs
• forbidden induced subgraph
• linear-time recognition