# On the Approximate Compressibility of Connected Vertex Cover

## TL;DR

This paper develops new polynomial size approximate kernelization schemes for the Connected Vertex Cover problem when parameterized by graph classes smaller than the solution size, such as cographs, bounded treewidth, and chordal graphs.

## Contribution

It introduces the first polynomial size approximate kernelization schemes for these specific graph class parameters, expanding the scope beyond solution size parameterization.

## Key findings

- First polynomial size approximate kernelization schemes for cographs, bounded treewidth, and chordal graphs.
- Improved understanding of approximate kernelization for Connected Vertex Cover.
- Extension of kernelization techniques to smaller graph parameters.

## Abstract

The Connected Vertex Cover problem, where the goal is to compute a minimum set of vertices in a given graph which forms a vertex cover and induces a connected subgraph, is a fundamental combinatorial problem and has received extensive attention in various subdomains of algorithmics. In the area of kernelization, it is known that this problem is unlikely to have efficient preprocessing algorithms, also known as polynomial kernelizations. However, it has been shown in a recent work of Lokshtanov et al. [STOC 2017] that if one considered an appropriate notion of approximate kernelization, then this problem parameterized by the solution size does admit an approximate polynomial kernelization. In fact, Lokhtanov et al. were able to obtain a polynomial size approximate kernelization scheme (PSAKS) for Connected Vertex Cover parameterized by the solution size. A PSAKS is essentially a preprocessing algorithm whose error can be made arbitrarily close to 0. In this paper we revisit this problem, and consider parameters that are strictly smaller than the size of the solution and obtain the first polynomial size approximate kernelization schemes for the Connected Vertex Cover problem when parameterized by the deletion distance of the input graph to the class of cographs, the class of bounded treewidth graphs, and the class of all chordal graphs.