TY - JOUR

T1 - Paths to trees and cacti

AU - Agrawal, Akanksha

AU - Kanesh, Lawqueen

AU - Saurabh, Saket

AU - Tale, Prafullkumar

N1 - Funding Information:
The research leading to these results has received funding from the European Research Council (ERC) via grant PARAPPROX, reference 306992 . The first and the third authors are supported by the PBC Program of Fellowships for Outstanding Post-doctoral Researchers from China and India (No. 5101479000 ); and Horizon 2020 Framework Programme , ERC Consolidator Grant LOPPRE (No. 819416 ), respectively.
Publisher Copyright:
© 2021 Elsevier B.V.

PY - 2021/3/8

Y1 - 2021/3/8

N2 - We know that TREE CONTRACTION does not admit a polynomial kernel unless NP ⊆ coNP/poly, while PATH CONTRACTION admits a kernel with O(k) vertices. The starting point of this article is the following natural questions: What is the structure of the family of paths that allows PATH CONTRACTION to admit a polynomial kernel? Apart from the size of the solution, what other additional parameters should we consider so we can design polynomial kernels for these basic contraction problems? To design polynomial kernels, we consider the family of trees with the bounded number of leaves (note that the family of paths are trees with at most two leaves). In particular, we study BOUNDED TREE CONTRACTION. Here, an input is a graph G, integers k and ℓ, and the goal is to decide whether, there is a subset F⊆E(G) of size at most k such that G/F is a tree with at most ℓ leaves. We design a kernel with O(kℓ) vertices and O(k2+kℓ) edges for this problem. We complement this result by giving kernelization lower bound. We also prove similar results for BOUNDED OUT-TREE CONTRACTION and BOUNDED CACTUS CONTRACTION.

AB - We know that TREE CONTRACTION does not admit a polynomial kernel unless NP ⊆ coNP/poly, while PATH CONTRACTION admits a kernel with O(k) vertices. The starting point of this article is the following natural questions: What is the structure of the family of paths that allows PATH CONTRACTION to admit a polynomial kernel? Apart from the size of the solution, what other additional parameters should we consider so we can design polynomial kernels for these basic contraction problems? To design polynomial kernels, we consider the family of trees with the bounded number of leaves (note that the family of paths are trees with at most two leaves). In particular, we study BOUNDED TREE CONTRACTION. Here, an input is a graph G, integers k and ℓ, and the goal is to decide whether, there is a subset F⊆E(G) of size at most k such that G/F is a tree with at most ℓ leaves. We design a kernel with O(kℓ) vertices and O(k2+kℓ) edges for this problem. We complement this result by giving kernelization lower bound. We also prove similar results for BOUNDED OUT-TREE CONTRACTION and BOUNDED CACTUS CONTRACTION.

KW - Graph contraction

KW - Kernel

KW - Kernel lower bound

UR - http://www.scopus.com/inward/record.url?scp=85099952678&partnerID=8YFLogxK

U2 - 10.1016/j.tcs.2021.01.033

DO - 10.1016/j.tcs.2021.01.033

M3 - Article

AN - SCOPUS:85099952678

VL - 860

SP - 98

EP - 116

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

ER -