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Maximum edge open packing admits polynomial-time algorithms on distance-hereditary graphs via twin-set decomposition.

2026-06-30 00:49 UTC pith:ZP6RZS6Q

load-bearing objection The paper adds polynomial algorithms for edge open packing on distance-hereditary and biconvex bipartite graphs plus an FPT result for chordal graphs, using standard structural decompositions.

arxiv 2606.28599 v1 pith:ZP6RZS6Q submitted 2026-06-26 math.CO cs.DM

Algorithms for the Maximum Edge Open Packing Problem

Sriram Bhyravarapu , Gautam K. Das , Kamal Santra This is my paper
classification math.CO cs.DM
keywords edge open packingdistance-hereditary graphsbiconvex bipartite graphschordal graphsparameterized complexitygraph algorithmspolynomial timeNP-hardness
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper studies the maximum edge open packing problem, where an edge open packing is a set of edges with no two joined by a third edge, generalizing induced matchings for conflict-free settings like wireless networks. It establishes that the size of a maximum such set can be computed in polynomial time on distance-hereditary graphs by exploiting their canonical decomposition based on twin-set interactions. The same polynomial-time result holds on biconvex bipartite graphs, a subclass of bipartite graphs where the problem is otherwise NP-hard on Eulerian instances. The work also gives a fixed-parameter tractable algorithm on chordal graphs when parameterized by the clique number.

Core claim

We give a polynomial-time algorithm for the problem in distance-hereditary graphs, exploiting their canonical decomposition via twin-set interactions. We further show that the problem remains polynomial-time solvable on biconvex bipartite graphs, thereby identifying a tractable subclass within bipartite graphs, in contrast to the known NP-hardness of the problem on Eulerian bipartite graphs. Finally, we initiate the parameterized complexity study of the problem and present a fixed-parameter tractable algorithm for chordal graphs, parameterized by the clique number ω, running in O(2^ω · poly(n)) time.

What carries the argument

Canonical decomposition of distance-hereditary graphs via twin-set interactions, which supports dynamic programming or reduction-based computation of the edge open packing number.

Load-bearing premise

The canonical decomposition of distance-hereditary graphs via twin-set interactions permits an efficient dynamic-programming or reduction-based computation of the edge open packing number.

What would settle it

A distance-hereditary graph on which the twin-set decomposition algorithm returns an edge open packing set whose size differs from the true maximum.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The edge open packing number is computable in polynomial time on all distance-hereditary graphs.
  • The problem is polynomial-time solvable on biconvex bipartite graphs even though it is NP-hard on Eulerian bipartite graphs.
  • The problem admits an FPT algorithm on chordal graphs when parameterized by clique number ω.
  • These results identify concrete graph classes where conflict-free edge selection can be decided efficiently.

Where Pith is reading between the lines

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

  • The twin-set approach may extend to other decomposable graph classes that arise in network interference models.
  • Small clique number could make the problem practical on chordal graphs that model certain hierarchical networks.
  • Similar parameterization by clique number might apply to related packing problems on chordal graphs.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 3 minor

Summary. The paper studies the Maximum Edge Open Packing Problem (a generalization of induced matching where edges conflict if joined by a third edge) and its computational complexity. It claims a polynomial-time algorithm for distance-hereditary graphs via their canonical twin-set decomposition, polynomial-time solvability on biconvex bipartite graphs (contrasting known NP-hardness on Eulerian bipartite graphs), and an FPT algorithm on chordal graphs parameterized by clique number ω running in O(2^ω · poly(n)) time.

Significance. If correct, the results identify new polynomial-time solvable classes for this packing problem using standard structural decompositions, provide a useful contrast within bipartite graphs, and begin the parameterized complexity analysis with a natural clique-number parameterization. The approach aligns with established techniques for distance-hereditary and chordal graphs.

minor comments (3)
  1. [Abstract] Abstract: the polynomial-time claims for distance-hereditary and biconvex graphs do not state explicit running times or high-level technique details (beyond 'exploiting canonical decomposition'), which would aid readers in assessing the contribution at a glance.
  2. [Introduction] The definition of edge open packing (no two edges joined by a third) is clear but an illustrative small example early in the introduction would help distinguish it from related notions like induced matching.
  3. [Abstract] Notation for the edge open packing number is introduced but could be used more consistently when stating the FPT running time bound in the abstract and conclusion.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our results on polynomial-time algorithms for distance-hereditary and biconvex bipartite graphs, as well as the FPT algorithm for chordal graphs, and for recommending minor revision.

Circularity Check

0 steps flagged

No significant circularity in algorithmic claims

full rationale

The paper presents new polynomial-time algorithms for the Maximum Edge Open Packing Problem on distance-hereditary graphs (via twin-set canonical decomposition) and biconvex bipartite graphs, plus an FPT algorithm on chordal graphs parameterized by clique number. These are standard structural algorithmic results relying on known graph decompositions and DP techniques; the abstract and context contain no equations, fitted parameters, self-definitional reductions, or load-bearing self-citations that collapse the claimed running times or tractability results back to the inputs by construction. The derivation chain is self-contained against external graph-theoretic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Results rest on standard structural properties of the named graph classes rather than new axioms or fitted parameters.

axioms (2)
  • domain assumption Distance-hereditary graphs admit a canonical decomposition based on twin-set interactions that supports efficient computation.
    Invoked for the polynomial-time algorithm on distance-hereditary graphs.
  • domain assumption Biconvex bipartite graphs form a tractable subclass for this packing problem.
    Used to establish polynomial-time solvability.

pith-pipeline@v0.9.1-grok · 5814 in / 1223 out tokens · 35197 ms · 2026-06-30T00:49:50.951201+00:00 · methodology

0 comments
read the original abstract

Packing problems form a central theme in graph theory, owing to their relevance in modeling conflict-free resource allocation, network design, and communication constraints. Motivated by applications in wireless networks where each device can participate in at most one communication at a time and simultaneous links must avoid interference we consider a generalization of induced matching known as \emph{edge open packing}. Two edges of a graph are said to conflict if a third edge connects one endpoint of each; an \emph{edge open packing set} is a set of edges containing no such conflicting pair. The largest cardinality of such a set is the \emph{edge open packing number} of a graph. In this work, we study the computational complexity of the Maximum Edge Open Packing Problem. We give a polynomial-time algorithm for the problem in \emph{distance-hereditary graphs}, exploiting their canonical decomposition via twin-set interactions. We further show that the problem remains polynomial-time solvable on \emph{biconvex bipartite graphs}, thereby identifying a tractable subclass within bipartite graphs, in contrast to the known NP-hardness of the problem on Eulerian bipartite graphs. Finally, we initiate the parameterized complexity study of the problem and present a fixed-parameter tractable algorithm for \emph{chordal graphs}, parameterized by the clique number $\omega$, running in $O(2^{\omega}\cdot\mathrm{poly}(n))$ time.

Figures

Figures reproduced from arXiv: 2606.28599 by Gautam K. Das, Kamal Santra, Sriram Bhyravarapu.

Figure 1
Figure 1. Figure 1: A path graph together with a tree decomposition of width 1. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: A distance-hereditary graph (left) and its decomposition tree (right). [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

discussion (0)

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Reference graph

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