Let be a fixed, finite family of graphs. In the -Deletion problem, the input is a graph G and a positive integer k, and the goal is to determine if there exists a set of at most k vertices whose deletion results in a graph that does not contain any graph of as a minor. The -Deletion problem encapsulates a large class of natural and interesting graph problems like Vertex Cover, Feedback Vertex Set, Treewidth- Deletion, Treedepth- Deletion, Pathwidth- Deletion, Outerplanar Deletion, Vertex Planarization and many more. We study the -Deletion problem from the kernelization perspective.
In a seminal work, Fomin, Lokshtanov, Misra & Saurabh [FOCS 2012] gave a polynomial kernel for this problem when the family F contains at least one planar graph. The asymptotic growth of the size of the kernel is not uniform with respect to the family : that is, the size of the kernel is , for some function f that depends only on . Also the size of the kernel depends on non-constructive constants.
Later Giannopoulou, Jansen, Lokshtanov & Saurabh [TALG 2017] showed that the non-uniformity in the kernel size bound of Fomin et al. is unavoidable as Treewidth- Deletion, cannot admit a kernel of size , for any , unless NP coNP/poly. On the other hand it was also shown that Treedepth- Deletion, admits a uniform kernel of size , showcasing that there are subclasses of where the asymptotic kernel sizes do not grow as a function of the family . This work leads to the natural question of determining classes of where the problem admits uniform polynomial kernels.
In this work, we show that if all the graphs in are connected and contains (a bipartite graph with 2 vertices on one side and p vertices on the other), then the problem admits a uniform kernel of size where the constants in the size bound are also constructive. The graph is one natural extension of the graph , where is a graph on two vertices and p parallel edges. The case when contains has been studied earlier and serves as (the only) other example where the problem admits a uniform polynomial kernel.
This is joint work with William Lochet.
The IBS discrete mathematics group welcomes Dr. Roohani Sharma, a new research fellow at the IBS Discrete Mathematics Group from May 1, 2025. She received her Ph.D. from the Institute of Mathematical Sciences, Chennai, India, under the supervision of Prof. Saket Saurabh. She is interested in parameterized complexity and kernelization. Previously, she was a researcher at the University of Bergen, Norway, and a Lise-Meitner Post-doctoral Fellow at the Max Planck Institute for Informatics in Germany.