Kernelization of Graph Hamiltonicity: Proper H-Graphs

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.