Contracting Graphs to Split Graphs and Threshold Graphs

TL;DR

This paper investigates the parameterized complexity of contracting graphs into split and threshold graphs, providing fixed-parameter tractable algorithms and kernelization hardness results for these problems.

Contribution

It introduces FPT algorithms for both Split Contraction and Threshold Contraction problems and establishes kernelization lower bounds, advancing understanding of their computational complexity.

Findings

FPT algorithm for Split Contraction problem.

FPT algorithm for Threshold Contraction on split graphs.

No polynomial kernels unless NP ⊆ coNP/poly.

Abstract

We study the parameterized complexity of Split Contraction and Threshold Contraction. In these problems we are given a graph G and an integer k and asked whether G can be modified into a split graph or a threshold graph, respectively, by contracting at most k edges. We present an FPT algorithm for Split Contraction, and prove that Threshold Contraction on split graphs, i.e., contracting an input split graph to a threshold graph, is FPT when parameterized by the number of contractions. To give a complete picture, we show that these two problems admit no polynomial kernels unless NP\subseteq coNP/poly.