eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-12-13
5:1
5:14
10.4230/LIPIcs.IPEC.2023.5
article
Difference Determines the Degree: Structural Kernelizations of Component Order Connectivity
Bhyravarapu, Sriram
1
Jana, Satyabrata
1
https://orcid.org/0000-0002-7046-0091
Saurabh, Saket
1
2
https://orcid.org/0000-0001-7847-6402
Sharma, Roohani
3
https://orcid.org/0000-0003-2212-1359
The Institute of Mathematical Sciences, HBNI, Chennai, India
University of Bergen, Norway
Max Planck Institute for Informatics, Saarland Informatics Campus, Saarbrücken, Germany
We consider the question of polynomial kernelization of a generalization of the classical Vertex Cover problem parameterized by a parameter that is provably smaller than the solution size. In particular, we focus on the c-Component Order Connectivity problem (c-COC) where given an undirected graph G and a non-negative integer t, the objective is to test whether there exists a set S of size at most t such that every component of G-S contains at most c vertices. Such a set S is called a c-coc set. It is known that c-COC admits a kernel with {O}(ct) vertices. Observe that for c = 1, this corresponds to the Vertex Cover problem.
We study the c-Component Order Connectivity problem parameterized by the size of a d-coc set (c-COC/d-COC), where c,d ∈ ℕ with c ≤ d. In particular, the input is an undirected graph G, a positive integer t and a set M of at most k vertices of G, such that the size of each connected component in G - M is at most d. The question is to find a set S of vertices of size at most t, such that the size of each connected component in G - S is at most c. In this paper, we give a kernel for c-COC/d-COC with O(k^{d-c+1}) vertices and O(k^{d-c+2}) edges. Our result exhibits that the difference in d and c, and not their absolute values, determines the exact degree of the polynomial in the kernel size.
When c = d = 1, the c-COC/d-COC problem is exactly the Vertex Cover problem parameterized by the solution size, which has a kernel with O(k) vertices and O(k²) edges, and this is asymptotically tight [Dell & Melkebeek, JACM 2014]. We also show that the dependence of d-c in the exponent of the kernel size cannot be avoided under reasonable complexity assumptions.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol285-ipec2023/LIPIcs.IPEC.2023.5/LIPIcs.IPEC.2023.5.pdf
Kernelization
Component Order Connectivity
Vertex Cover
Structural Parameterizations