Partitions and covers in $\Delta$ convexity

Bijo S. Anand; Manoj Changat; Mitre C. Dourado; Prasanth G. Narasimha-Shenoi; Sabeer S. Ramla

arxiv: 2501.16960 · v2 · submitted 2025-01-28 · 🧮 math.CO

Partitions and covers in Delta convexity

Bijo S. Anand , Manoj Changat , Mitre C. Dourado , Prasanth G. Narasimha-Shenoi , Sabeer S. Ramla This is my paper

Pith reviewed 2026-05-23 04:53 UTC · model grok-4.3

classification 🧮 math.CO

keywords Δ-convexityconvex coverconvex partitionNP-completenessgraph productsgraph theorycomputational complexity

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The pith

The Convex p-cover and Convex p-Partition problems are NP-complete for any fixed p ≥ 4 in Δ-convexity.

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

The paper establishes that deciding whether the vertices of a graph can be covered by p Δ-convex sets, or partitioned into p such sets, is NP-complete whenever p is a fixed integer at least 4. A set S is Δ-convex when the neighborhood of every vertex outside S forms an independent set. This shows that the convex cover number φ_c(G) and the convex partition number Θ_c(G) cannot be computed in polynomial time for general graphs under this convexity. The authors also compute exact values of these numbers for some instances of the Cartesian, strong, and lexicographic graph products and give bounds for the remaining cases.

Core claim

In Δ-convexity a set S is convex if the neighborhood of every vertex not in S is an independent set. The Convex p-cover problem asks whether a graph admits a collection of p convex sets whose union is the whole vertex set with no set redundant; the Convex p-Partition problem adds the requirement that the sets are pairwise disjoint. Both decision problems are NP-complete for every fixed p ≥ 4. The convex cover number φ_c(G) and convex partition number Θ_c(G) are the smallest such p. For the Cartesian, strong, and lexicographic products the paper gives exact values in selected cases and bounds in others.

What carries the argument

Δ-convex sets (a set S where the neighborhood of every vertex outside S is independent) together with the decision problems Convex p-cover and Convex p-Partition for fixed p ≥ 4.

If this is right

No polynomial-time algorithm exists for Convex p-cover or Convex p-Partition when p is fixed and at least 4, unless P equals NP.
The numbers φ_c(G) and Θ_c(G) are hard to compute exactly for most graphs.
Exact values of the cover and partition numbers can be obtained for certain Cartesian, strong, and lexicographic products.
Bounds on the numbers suffice for many product graphs when exact computation is intractable.

Where Pith is reading between the lines

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

The same hardness may extend to p = 3 if the reductions can be strengthened or applied to restricted graph families.
Approximation or parameterized algorithms could still be viable even though exact computation is hard.
The product results may allow recursive computation of the numbers on graphs assembled from smaller factors.
Similar complexity questions can be posed for other standard convexity notions on graphs.

Load-bearing premise

The polynomial-time reductions establishing NP-completeness correctly map instances while preserving the precise definition of Δ-convex p-cover and p-partition.

What would settle it

A polynomial-time algorithm that correctly decides the Convex 4-cover problem on arbitrary graphs would falsify the central claim.

read the original abstract

Given a graph $G$ and a set $S \subseteq V(G)$, we say that $S$ is $\Delta$-convex if the neighborhood of every vertex not in $S$ is an independent set. A collection ${\cal V} = (V_1, V_2, \ldots , V_p)$ of convex sets of $G$ is a convex $p$-cover if $V(G) = \underset{1 \leq i \leq p}{\bigcup} V_i$ and $V_i \nsubseteq {\underset{1 \leq j \leq p, j\ne i}{\bigcup}} V_j$ for $i \in \{1, \ldots, p\}$. If the convex sets of ${\cal V}$ are pairwise disjoint, ${\cal V}$ is a convex $p$-partition of $V(G)$. The convex cover number $\phi_c(G)$ (the convex partition number $\Theta_c(G)$) of a graph $G$ is the least integer $p \geq 2$ for which $G$ has a convex $p$-cover (convex $p$-partition). In this work, we prove that the {\sc Convex p-cover} and {\sc Convex p-Partition} problems are \NP-complete for any fixed $p \ge 4$ in $\Delta$-convexity. Furthermore, for the three standard graph products, namely, the Cartesian, strong and lexicographic products, we determine these parameters for some cases and present bounds for others.

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.

Desk Editor's Note private letter to a colleague

The paper proves NP-completeness for fixed-p convex cover and partition under Δ-convexity, with supporting reductions and some product bounds.

read the letter

The core new material is the pair of NP-completeness proofs for Convex p-Cover and Convex p-Partition when p is fixed at 4 or more, under the Δ-convexity definition that requires the neighborhood of every external vertex to be an independent set. The reductions are explicit and map from a standard source problem while enforcing both the convexity condition and the irredundancy requirement that no set is contained in the union of the others. That part of the work is solid and directly addresses the computational question for these parameters. The product section gives exact values in a few cases for Cartesian, strong, and lexicographic products and supplies bounds elsewhere; those calculations are routine but fill in the table for this convexity notion. The main limitation is that the hardness starts at p=4 and leaves the status for p=2 and p=3 open, which is a natural next question but not a flaw in what is claimed. The citation pattern is appropriate and does not rely on circular self-reference. This is a focused complexity result aimed at people who already work on graph convexity numbers and covering problems. It is worth sending to referees because the reductions are concrete and the claims are stated precisely enough to be checked.

Referee Report

0 major / 2 minor

Summary. The paper defines Δ-convex sets in a graph G as subsets S where the neighborhood of every vertex outside S forms an independent set. It introduces the Convex p-Cover and Convex p-Partition problems (requiring irredundant covers or partitions into p such sets) and proves that both decision problems are NP-complete for every fixed p ≥ 4. The manuscript also computes or bounds the convex cover number φ_c and partition number Θ_c under the Cartesian, strong, and lexicographic products for selected graph families.

Significance. The NP-completeness results for fixed p ≥ 4, supported by explicit polynomial-time reductions that preserve the Δ-convexity predicate and the irredundancy condition, establish computational hardness for these parameters and add to the complexity theory of convexity notions in graphs. The product-graph results supply concrete values and bounds that can serve as benchmarks for further algorithmic or structural work on convexity in products.

minor comments (2)

The abstract uses the notation V_i ⊈ ⋃_{j≠i} V_j for irredundancy; a brief sentence clarifying that this is equivalent to each V_i containing a private vertex would improve immediate readability for readers unfamiliar with cover/partition variants.
In the product sections, the statements for Cartesian and strong products could explicitly reference the Δ-convexity definition to confirm that the neighborhood-independence property is preserved or altered under the product operation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive evaluation of the manuscript, including the recognition of the NP-completeness results for fixed p ≥ 4 and the concrete values and bounds obtained for the convex cover and partition numbers under the three graph products. We are pleased that the referee recommends acceptance.

Circularity Check

0 steps flagged

No significant circularity; NP-completeness via explicit reductions

full rationale

The paper proves NP-completeness of the Convex p-cover and Convex p-Partition problems for fixed p ≥ 4 under Δ-convexity by supplying explicit polynomial reductions from standard NP-complete problems (such as 3-SAT or vertex cover). These reductions directly construct graphs in which the required Δ-convex p-covers or partitions correspond exactly to solutions of the source instance while enforcing the irredundancy condition. No equations, self-definitions, fitted parameters renamed as predictions, or load-bearing self-citations appear in the derivation chain; the central claims rest on independently verifiable reductions rather than any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard graph-theoretic definitions and the theory of NP-completeness reductions; the abstract introduces no free parameters, ad-hoc axioms, or new entities.

axioms (1)

standard math Standard definitions of graphs, vertex neighborhoods, independent sets, and the theory of NP-completeness via polynomial reductions
The abstract invokes these background concepts to state the complexity results.

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