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Polynomial Kernel and Incompressibility for Prison-Free Edge Deletion and Completion

Séhane Bel Houari-Durand, Eduard Eiben, Magnus Wahlström

TL;DR

This work studies parameterized graph modification with respect to $H$-free constraints, focusing on the prison graph. It proves a sharp dichotomy: Prison-Free Edge Deletion admits a polynomial kernel, while Prison-Free Edge Completion is incompressible unless the polynomial hierarchy collapses, highlighting a fundamental difference between deletion and completion variants. Central to the results is a structural characterization of prison-free graphs via the complete-multipartite decomposition $cmd_4(G)$, enabling a sunflower-based modulator and targeted reductions. The combination of a structural graph theorem, gap-hardness, cross-composition, and a carefully designed kernelization pipeline demonstrates how graph structure drives kernelizability in $H$-free edge modification problems and settles key conjectures in the area.

Abstract

Given a graph $G$ and an integer $k$, the $H$-free Edge Deletion problem asks whether there exists a set of at most $k$ edges of $G$ whose deletion makes $G$ free of induced copies of $H$. Significant attention has been given to the kernelizability aspects of this problem -- i.e., for which graphs $H$ does the problem admit an "efficient preprocessing" procedure, known as a polynomial kernelization, where an instance $I$ of the problem with parameter $k$ is reduced to an equivalent instance $I'$ whose size and parameter value are bounded polynomially in $k$? Although such routines are known for many graphs $H$ where the class of $H$-free graphs has significant restricted structure, it is also clear that for most graphs $H$ the problem is incompressible, i.e., admits no polynomial kernelization parameterized by $k$ unless the polynomial hierarchy collapses. These results led Marx and Sandeep to the conjecture that $H$-free Edge Deletion is incompressible for any graph $H$ with at least five vertices, unless $H$ is complete or has at most one edge (JCSS 2022). This conjecture was reduced to the incompressibility of $H$-free Edge Deletion for a finite list of graphs $H$. We consider one of these graphs, which we dub the prison, and show that Prison-Free Edge Deletion has a polynomial kernel, refuting the conjecture. On the other hand, the same problem for the complement of the prison is incompressible.

Polynomial Kernel and Incompressibility for Prison-Free Edge Deletion and Completion

TL;DR

This work studies parameterized graph modification with respect to -free constraints, focusing on the prison graph. It proves a sharp dichotomy: Prison-Free Edge Deletion admits a polynomial kernel, while Prison-Free Edge Completion is incompressible unless the polynomial hierarchy collapses, highlighting a fundamental difference between deletion and completion variants. Central to the results is a structural characterization of prison-free graphs via the complete-multipartite decomposition , enabling a sunflower-based modulator and targeted reductions. The combination of a structural graph theorem, gap-hardness, cross-composition, and a carefully designed kernelization pipeline demonstrates how graph structure drives kernelizability in -free edge modification problems and settles key conjectures in the area.

Abstract

Given a graph and an integer , the -free Edge Deletion problem asks whether there exists a set of at most edges of whose deletion makes free of induced copies of . Significant attention has been given to the kernelizability aspects of this problem -- i.e., for which graphs does the problem admit an "efficient preprocessing" procedure, known as a polynomial kernelization, where an instance of the problem with parameter is reduced to an equivalent instance whose size and parameter value are bounded polynomially in ? Although such routines are known for many graphs where the class of -free graphs has significant restricted structure, it is also clear that for most graphs the problem is incompressible, i.e., admits no polynomial kernelization parameterized by unless the polynomial hierarchy collapses. These results led Marx and Sandeep to the conjecture that -free Edge Deletion is incompressible for any graph with at least five vertices, unless is complete or has at most one edge (JCSS 2022). This conjecture was reduced to the incompressibility of -free Edge Deletion for a finite list of graphs . We consider one of these graphs, which we dub the prison, and show that Prison-Free Edge Deletion has a polynomial kernel, refuting the conjecture. On the other hand, the same problem for the complement of the prison is incompressible.

Paper Structure

This paper contains 11 sections, 31 theorems, 2 figures.

Table of Contents

  1. Introduction
  2. Our results
  3. Structure of prison-free graphs
  4. Incompressibility of Prison-free Edge Completion
  5. Initial observations and support gadgets
  6. NP-hardness of Gap Prison-free Edge Completion
  7. Compositionality of Prison-Free Edge Completion
  8. Polynomial kernel for Prison-free Edge Deletion
  9. Finding a Small Vertex Modulator
  10. Consequences on the Structure of $G[\bar{S}]$
  11. Marking important vertices

Key Result

Theorem 2

A graph $G=(V,E)$ is prison-free if and only if the following holds: Let $F \subseteq V$ be an inclusion-wise maximal set such that $G[F]$ is complete multipartite with at least 4 parts, and let $v \in V \setminus F$. Then $N(v)$ intersects at most one part of $F$.

Figures (2)

  • Figure 1: Three graphs: The prison, the paw, and the diamond (as subgraphs of the prison).
  • Figure 2: Gadgets for Section \ref{['sec:lb']}. Dotted lines are forbidden edges; dashed lines are named "gadget-edges" with special semantics.

Theorems & Definitions (34)

  • Conjecture 1: Conjecture 2 of 1
  • Theorem 2
  • Corollary 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Lemma 6
  • Theorem 6
  • Corollary 6
  • Proposition 7
  • ...and 24 more