@article{f9bae5b10aba4d0f853bd582f608025f,
title = "KERNELIZATION OF GRAPH HAMILTONICITY: Proper H-Graphs",
abstract = "We obtain new polynomial kernels and compression algorithms for Path Cover and Cycle Cover, the well-known generalizations of the classical Hamiltonian Path and Hamiltonian Cycle problems. Our choice of parameterization is strongly influenced by the work of Bir\'o, Hujter, and Tuza, who in 1992 introduced H-graphs, intersection graphs of connected subgraphs of a subdivision of a fixed (multi-)graph H. In this work, we turn to proper H-graphs, where the containment relationship between the representations of the vertices is forbidden. As the treewidth of a graph measures how similar the graph is to a tree, the size of graph H is the parameter measuring the closeness of the graph to a proper interval graph. We prove the following results. Path Cover admits a kernel of size O (parallel to H parallel to(8)), where parallel to H parallel to is the size of graph H. In other words, we design an algorithm that for an n-vertex graph G and integer k \geq 1, in time polynomial in n and parallel to H parallel to, outputs a graph G\prime of size \scrO (parallel to H parallel to(8)) and k\prime \leq | V (G' such that the vertex set of G is coverable by k vertex-disjoint paths if and only if the vertex set of G' is coverable by k' vertex-disjoint paths. Hamiltonian Cycle admits a kernel of size O (parallel to H parallel to(8)). Cycle Cover admits a polynomial kernel. We prove it by providing a compression of size O (parallel to H parallel to(10)) into another NP-complete problem, namely, Prize Collecting Cycle Cover, that is, we design an algorithm that, in time polynomial in n and parallel to H parallel to, outputs an equivalent instance of Prize Collecting Cycle Cover of sizeO (parallel to H parallel to(10)). In all our algorithms we assume that a proper H-decomposition is given as a part of the input.",
keywords = "cycle cover, path cover, proper H-graphs, kernelization, LOG N) ALGORITHM, INTERVAL-GRAPHS, CIRCUITS, PATHS",
author = "Steven Chaplick and Fedor Fomin and Golovach, {Petr A.} and Dusan Knop and Peter Zeman",
year = "2021",
doi = "10.1137/19M1299001",
language = "English",
volume = "35",
pages = "840--892",
journal = "Siam Journal on Discrete Mathematics",
issn = "0895-4801",
publisher = "SIAM Publications",
number = "2",
}