# Kernelization of Graph Hamiltonicity - Proper H-Graphs

Steven Chaplick, Fedor V. Fomin, Petr A. Golovach, Dusan Knop*, Peter Zeman

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference article in proceedingAcademicpeer-review

1 Citation (Web of Science)

## 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 Biro, 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)), that is, we design an algorithm that for an n-vertex graph G and an integer k >= 1, in time polynomial in n and parallel to H parallel to, outputs a graph G' of size O(parallel to H parallel to(8)) and k'

- CYCLE COVER admits a compression of size O(parallel to H parallel to(10)) into another problem, called 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 size O(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.

Original language English Algorithms and Data Structures. WADS 2019 Z. Friggstad, JR Sack, M. Salavatipour Springer, Cham 296-310 15 978-3-030-24766-9 978-3-030-24765-2 https://doi.org/10.1007/978-3-030-24766-9_22 Published - 2019 Yes

### Publication series

Series Lecture Notes in Computer Science 11646 0302-9743

## Keywords

• CIRCUITS
• Cycle Cover
• INTERVAL-GRAPHS
• Kernelization
• LOG N) ALGORITHM
• PATH
• Path Cover
• Proper H-graphs
• ### KERNELIZATION OF GRAPH HAMILTONICITY: Proper H-Graphs

Chaplick, S., Fomin, F., Golovach, P. A., Knop, D. & Zeman, P., 2021, 35, 2, p. 840-892 53 p.