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 \scrO (\| H\| 8 ), where \| H\| 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 \| H\| , outputs a graph G \prime of size \scrO (\| H\| 8 ) and k \prime \leq | V (G \prime )| such that the vertex set of G is coverable by k vertex-disjoint paths if and only if the vertex set of G \prime is coverable by k \prime vertex-disjoint paths. Hamiltonian Cycle admits a kernel of size \scrO (\| H\| 8 ). Cycle Cover admits a polynomial kernel. We prove it by providing a compression of size \scrO (\| H\| 10 ) into another NP-complete problem, namely, Prize Collecting Cycle Cover, that is, we design an algorithm that, in time polynomial in n and \| H\| , outputs an equivalent instance of Prize Collecting Cycle Cover of size \scrO (\| H\| 10 ). In all our algorithms we assume that a proper H-decomposition is given as a part of the input.

Treewidth versus clique number. II. Tree-independence number

“…Intersection graphs of connected subgraphs of graphs with treewidth at most t were studied in 1990 by Scheffler [104] and in 1998 by Bodlaender, Gustedt, and Telle [16]. They include chordal graphs (for which t = 1, since every chordal graphs is the intersection graph of subtrees in a tree [31,74,116]) and circular-arc graphs, that is, intersection graphs of circular arcs on a circle (for which t = 2), as well as H-graphs, that is, the intersection graphs of connected subgraphs of a subdivision of a fixed multigraph H, introduced in 1992 by Bíró, Hujter, and Tuza [14] and studied more recently in a number of papers [36][37][38]67]. Classes of graphs in which all minimal separators are of bounded size were studied in 1999 by Skodinis [107].…”

Intersection Graphs of Non-crossing Paths

“…If x is a junction, then (i) x has degree 3 and (ii) x's neighbors are terminals. 3. If x has degree four or more, then x is a terminal.…”

“…Note: all ∃R-hard problems are NP-hard, see[27] for an introduction to ∃R 2. Usually defined as having no interval strictly contained within any other 3. A graph is chordal when it has no induced cycles of length four or more.…”

Recognizing Proper Tree-Graphs

“…The classes of H-graphs have seen renewed interest in recent years concerning their structure and recognition [8], relation to other graph parameters [8,12], and primarily regarding the computational complexity of standard algorithmic problems when parameterized by the size of H [1,7,8,9,12,17,18]. Of particular relevance to our paper is the work on Hamiltonicity problems [7] as it introduces proper H-graphs, which are to proper interval graphs as H-graphs are to interval graphs. Namely, for a graph G, a subdivision H sub of H properly represents G when H sub represents G using models…”

“…In particular, on proper H-graphs polynomial size kernels (in the size of H) were developed for various Hamiltonicity problems [7], but the recognition problems were left open.…”