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

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.

• The paper demonstrates that, for distance-hereditary graphs, balancedness, the hereditary clique-Helly property, and the absence of induced ¯3K2 are equivalent.
• It introduces a certificate-based recursive algorithm that leverages OVE-tree decompositions to detect forbidden subgraphs in linear time.
• The findings provide both theoretical insights and practical applications for optimizing combinatorial problems in structured graph classes.

Characterization and Linear-Time Recognition of Balanced Distance-Hereditary Graphs

Introduction

The study focuses on the structural and algorithmic properties of balanced graphs within the class of distance-hereditary graphs. Balanced graphs are defined via the balancedness of their clique-matrix, with the exclusion of odd square submatrices having exactly two $1$’s per row and column. While a full characterization of balanced graphs by minimal forbidden induced subgraphs is still open, the authors present a complete structural and algorithmic understanding in the context of distance-hereditary graphs.

The key contributions are twofold. First, the authors show that, for distance-hereditary graphs, being balanced is equivalent to being hereditary clique-Helly, and these are exactly the graphs forbidding $\overline{3K_2}$ as an induced subgraph. Second, the paper provides a linear-time recognition algorithm—a substantial improvement over generic polynomial-time methods for recognizing balancedness.

Background and Preliminaries

Balanced graphs were introduced by Berge and Chvátal, and their clique-matrix definition implies perfectness (no odd holes/antiholes) and clique-perfectness. Earlier, Bonomo et al. had shown that balancedness in general graphs can be checked via the absence of induced extended odd suns, but these are not characterized by minimal forbidden induced subgraphs. Moreover, the only polynomial recognition method relied on matrix results, being $O(m^9 + n)$.

Distance-hereditary graphs, introduced by Howorka, admit several structural characterizations: they can be built from $K_1$ iteratively via pendant, true twin, and false twin additions, they are characterized by the absence of induced house, gem, domino, or holes of length ≥5, and, notably, their structure is reflected in so-called one-vertex-extension trees (OVE-trees).

The hereditary clique-Helly property ensures that all induced subgraphs have the Helly property for maximal cliques. Prisner’s result implies hereditary clique-Helly graphs are characterized by a short list of forbidden induced subgraphs. Within cographs, balancedness coincides with the hereditary clique-Helly property and can be captured by forbidding $\overline{3K_2}$.

Main Structural Theorem

The central claim is the equivalence, within distance-hereditary graphs, of three properties:

1. Balancedness (clique-matrix property).
2. Hereditary clique-Helly property.
3. Absence of induced $\overline{3K_2}$.

The main theorem: For any distance-hereditary graph $G$, $G$ is balanced if and only if it is hereditary clique-Helly if and only if $G$ does not contain $\overline{3K_2}$ as an induced subgraph (2607.00730).

This result is nontrivial, as, for general graphs, balancedness is not closed under induced subgraphs nor characterized by a single forbidden induced subgraph. The authors provide a full inductive characterization using OVE-tree decompositions, leveraging the building operations (pendant, true/false twin) and systematically tracking the preservation of balancedness and hereditary clique-Helly status.

Algorithmic Contribution

Given an OVE-tree for the input graph (obtainable in linear time for distance-hereditary graphs [Chang et al., 1997]), the authors design a certificate-based recursive algorithm for balancedness detection. The certificates monitor the presence of required forbidden configurations (e.g., induced $\overline{3K_2}$0 in a twin set, induced $\overline{3K_2}$1 with the universal vertex outside the twin set, induced $\overline{3K_2}$2 crossing construction steps).

The recognition algorithm operates as follows:

• Use a linear-time recognition procedure to produce an OVE-tree for the input graph.
• Traverse this tree in a bottom-up fashion to compute, for each subtree, the presence or absence of forbidden subgraphs.
• If any induced $\overline{3K_2}$3 is detected, output the corresponding certificate; else, conclude balancedness.

The runtime is linear in the size of the input graph, which is optimal for this class.

Theoretical Implications

This work contributes fundamentally to structural graph theory, revealing that, unlike the general case, balancedness is robust against the hereditary closure operation in distance-hereditary graphs. This shows a sharp contrast with perfectness and clique-perfectness, where clique-perfect graphs still lack a forbidden subgraph characterization.

The result situates the hereditary clique-Helly property at the intersection of several classes: it extends the cograph result and demonstrates that, for the much broader class of distance-hereditary graphs, minimal forbidden induced subgraph characterization is achieved by a single, small configuration.

From the perspective of algorithmic graph theory, the authors' synthesis of OVE-trees and certificate propagation illustrates how structural decompositions compatible with hereditary properties can yield efficient recognition algorithms—opening the path for similar analyses in other well-structured graph classes.

Practical Relevance and Future Prospects

Practically, recognizing balanced, clique-Helly, and forbidden-$\overline{3K_2}$4 graphs efficiently supports various applications in combinatorial optimization, where clique intersection properties impact algorithmic tractability. For example, scheduling, resource allocation, and matrix optimization can exploit these properties for efficient solutions.

Future directions include:

• Extending minimal forbidden induced subgraph characterizations to other hereditary subclasses containing or extending distance-hereditary graphs.
• Investigating whether similar reductions can be established for the recognition of related properties, e.g., clique-perfectness or more general matrix-balancedness, in other perfect graph classes.
• Analyzing the parameterized complexity of recognition for near-extensions of distance-hereditary graphs and for looser structural decompositions.

Conclusion

The authors establish that, for distance-hereditary graphs, balancedness, the hereditary clique-Helly property, and the absence of induced $\overline{3K_2}$5 are all equivalent properties. Moreover, they provide a linear-time recognition algorithm, yielding both practical and theoretical progress in the study of structurally restricted graph classes. The results clarify the relationships between fundamental intersection and matrix properties within a significant family of perfect graphs, while setting a methodological standard for further research into hereditary graph properties and their algorithmic exploitation.

Reference:

"Characterization and linear-time recognition of balanced distance-hereditary graphs" (2607.00730)