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arXiv · 2607.00730

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

Lucía Busolini·Guillermo Durán·Martín D. Safe
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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.

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Lucía Busolini, Guillermo Durán, Martín D. Safe. 2026-07-01. Characterization and linear-time recognition of balanced distance-hereditary graphs. https://arxiv.org/abs/2607.00730

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