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arxiv: 2607.00730 · v1 · pith:XFLO545Inew · submitted 2026-07-01 · 🧮 math.CO · cs.DM

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

Luc\'ia Busolini , Guillermo Dur\'an , Mart\'in D. Safe This is my paper

Pith reviewed 2026-07-02 10:56 UTC · model grok-4.3

classification 🧮 math.CO cs.DM
keywords balanced graphsdistance-hereditary graphshereditary clique-Helly graphsforbidden induced subgraphslinear-time recognitionclique matrix
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The pith

Within distance-hereditary graphs, balanced graphs are exactly the hereditary clique-Helly graphs and are characterized by the single forbidden induced subgraph ¯{3K_2}.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that for distance-hereditary graphs, being balanced is equivalent to being hereditary clique-Helly. This means they can be characterized by forbidding one specific induced subgraph, the complement of three disjoint edges. This equivalence is used to create a linear-time algorithm for recognizing whether a distance-hereditary graph is balanced. A sympathetic reader would care because it simplifies the recognition problem in a well-studied class of graphs. The result connects balancedness, defined via the clique matrix, with clique-Helly properties.

Core claim

Balanced distance-hereditary graphs are exactly the hereditary clique-Helly graphs. Equivalently, they are the graphs without an induced ¯{3K_2}. From this characterization, there is an explicit linear-time algorithm that decides if a distance-hereditary graph is balanced and finds an induced ¯{3K_2} if it is not.

What carries the argument

The equivalence of balancedness and the hereditary clique-Helly property within distance-hereditary graphs, which reduces the problem to checking for the absence of the induced subgraph ¯{3K_2}.

If this is right

  • Balancedness can be decided in linear time for any distance-hereditary graph.
  • The algorithm explicitly returns an induced ¯{3K_2} when the input is not balanced.
  • The general open problem of characterizing all balanced graphs by forbidden subgraphs is solved inside the distance-hereditary class.
  • Recognition reduces to a single forbidden-subgraph test rather than checking the clique matrix directly.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The structural restrictions of distance-hereditary graphs may make other matrix properties or Helly-type conditions easier to decide than in arbitrary graphs.
  • Similar single-forbidden-subgraph results could be sought for balancedness inside other hereditary classes such as chordal or perfect graphs.
  • The linear-time procedure could be implemented directly on the standard linear-time recognition algorithms already known for distance-hereditary graphs.

Load-bearing premise

The result relies on the known characterization of hereditary clique-Helly graphs as exactly those without induced ¯{3K_2} remaining valid when restricted to distance-hereditary graphs.

What would settle it

A distance-hereditary graph that contains an induced ¯{3K_2} yet remains balanced, or a hereditary clique-Helly distance-hereditary graph that fails to be balanced.

Figures

Figures reproduced from arXiv: 2607.00730 by Guillermo Dur\'an, Luc\'ia Busolini, Mart\'in D. Safe.

Figure 1
Figure 1. Figure 1: An extended odd sun that is not a minimal induced subgraph for the class of balanced [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The graph 3K2. Balanced graphs were characterized by minimal forbidden induced subgraphs when restricted to diamond-free graphs [1], some subclasses of circular-arc graphs [10], paw-free graphs [11], P4- tidy graphs [11], claw-free graphs [13], complements of bipartite graphs, line graphs of multi￾graphs, and complements of line graphs of multigraphs [9]. Balanced graphs can be recognized in polynomial tim… view at source ↗
Figure 3
Figure 3. Figure 3: The house, the gem, and the domino. (iii) G contains no induced house, gem, or domino (shown in [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper proves that, restricted to the class of distance-hereditary graphs, the balanced graphs are precisely the hereditary clique-Helly graphs. As a consequence, balanced distance-hereditary graphs are characterized by the single forbidden induced subgraph ¯{3K_2}. The authors also derive an explicit linear-time recognition algorithm for balancedness in this class that returns an induced ¯{3K_2} on negative instances.

Significance. If the central equivalence holds, the result supplies a simple forbidden-subgraph characterization and a practical linear-time algorithm for a nontrivial subclass of balanced graphs. The work correctly invokes the known external fact that hereditary clique-Helly graphs are exactly the ¯{3K_2}-free graphs and shows that this fact specializes directly once the balanced/hereditary-clique-Helly equivalence is established inside distance-hereditary graphs. The explicit linear-time algorithm is a concrete algorithmic contribution.

minor comments (3)
  1. The abstract states the equivalence but does not name the section containing the proof; adding a forward reference would improve readability.
  2. Notation for the complement of 3K_2 is introduced as ¯{3K_2} in the abstract; ensure the same symbol is used consistently in all figures and statements of the main theorem.
  3. The complexity analysis of the recognition algorithm is asserted to be linear but is not cross-referenced to a specific lemma or theorem number in the abstract; a pointer would help readers locate the running-time argument.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation to accept the manuscript.

Circularity Check

0 steps flagged

No circularity in derivation chain

full rationale

The paper's central result is an explicit proof establishing equivalence between balanced and hereditary clique-Helly properties restricted to distance-hereditary graphs, followed by invocation of the independent external characterization that hereditary clique-Helly graphs are exactly the ¯{3K_2}-free graphs. This external fact restricts immediately to the subclass without requiring additional conditions or self-referential definitions. No load-bearing step reduces by construction to a fitted parameter, self-citation chain, or ansatz smuggled from the authors' prior work; the derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result depends on standard definitions of balanced graphs, distance-hereditary graphs, and hereditary clique-Helly graphs, plus the known forbidden-subgraph characterization of the latter class. No free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (2)
  • standard math A graph is balanced iff its clique-matrix contains no square submatrix of odd order with exactly two 1s in each row and column.
    Invoked in the opening sentence of the abstract as the definition of balanced graphs.
  • domain assumption Hereditary clique-Helly graphs are exactly the graphs without induced ¯{3K_2}.
    Used to equate the two properties inside distance-hereditary graphs; this external fact is required for the single-forbidden-subgraph claim.

pith-pipeline@v0.9.1-grok · 5673 in / 1469 out tokens · 25290 ms · 2026-07-02T10:56:01.441998+00:00 · methodology

discussion (0)

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