eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-12-04
10:1
10:15
10.4230/LIPIcs.IPEC.2020.10
article
A Polynomial Kernel for Paw-Free Editing
Eiben, Eduard
1
https://orcid.org/0000-0003-2628-3435
Lochet, William
2
Saurabh, Saket
3
2
Department of Computer Science, Royal Holloway, University of London, Egham, UK
Department of Informatics, University of Bergen, Norway
Institute of Mathematical Sciences, Chennai, India
For a fixed graph H, the H-free Edge Editing problem asks whether we can modify a given graph G by adding or deleting at most k edges such that the resulting graph does not contain H as an induced subgraph. The problem is known to be NP-complete for all fixed H with at least 3 vertices and it admits a 2^O(k)n^O(1) algorithm. Cai and Cai [Algorithmica (2015) 71:731–757] showed that, assuming coNP ⊈ NP/poly, H-free Edge Editing does not admit a polynomial kernel whenever H or its complement is a path or a cycle with at least 4 edges or a 3-connected graph with at least one edge missing. Based on their result, very recently Marx and Sandeep [ESA 2020] conjectured that if H is a graph with at least 5 vertices, then H-free Edge Editing has a polynomial kernel if and only if H is a complete or empty graph, unless coNP ⊆ NP/poly. Furthermore they gave a list of 9 graphs, each with five vertices, such that if H-free Edge Editing for these graphs does not admit a polynomial kernel, then the conjecture is true. Therefore, resolving the kernelization of H-free Edge Editing for graphs H with 4 and 5 vertices plays a crucial role in obtaining a complete dichotomy for this problem. In this paper, we positively answer the question of compressibility for one of the last two unresolved graphs H on 4 vertices. Namely, we give the first polynomial kernel for Paw-free Edge Editing with O(k⁶) vertices.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol180-ipec2020/LIPIcs.IPEC.2020.10/LIPIcs.IPEC.2020.10.pdf
Kernelization
Paw-free graph
H-free editing
graph modification problem