Tight Kernel Bounds for Problems on Graphs with Small Degeneracy

TL;DR

This paper investigates kernelization bounds for problems on d-degenerate graphs, showing that existing upper bounds are nearly optimal for various natural problems, thereby advancing understanding of kernelization limits in this graph class.

Contribution

It establishes tight bounds for kernelization on d-degenerate graphs, generalizing known results from specific graph classes and demonstrating the near-optimality of existing bounds.

Findings

Kernelization bounds are essentially tight for several problems on d-degenerate graphs.

D-degenerate graphs include many well-studied graph classes like planar and H-minor free graphs.

The results generalize and unify kernelization bounds across various graph classes.

Abstract

In this paper we consider kernelization for problems on d-degenerate graphs, i.e. graphs such that any subgraph contains a vertex of degree at most $d$. This graph class generalizes many classes of graphs for which effective kernelization is known to exist, e.g. planar graphs, H-minor free graphs, and H-topological-minor free graphs. We show that for several natural problems on d-degenerate graphs the best known kernelization upper bounds are essentially tight.