Three ways to cover a graph

Discrete Mathematics 05/2012; 339(2). DOI: 10.1016/j.disc.2015.10.023
Source: arXiv


We consider the problem of covering a host graph $G$ with several graphs from
a fixed template class ${T}$. The classical covering number of $G$ with respect
to ${T}$ is the minimum number of template graphs needed to cover the edges of
$G$. We introduce two new parameters: the local and the folded covering number.
Each parameter measures how far $G$ is from the template class in a different
way. Whereas the folded covering number has been investigated thoroughly for
some template classes, e.g., interval graphs and planar graphs, the local
covering number was given only little attention.
We provide new bounds on each covering number w.r.t. the following template
classes: linear forests, star forests, caterpillar forests, and interval
graphs. The classical graph parameters turning up this way are interval-number,
track-number, and linear-, star-, and caterpillar arboricity. As host graphs we
consider graphs of bounded degeneracy, bounded degree, or bounded (simple)
tree-width, as well as, outerplanar, planar bipartite and planar graphs. For
several pairs of a host class and a template class we determine the maximum
(local, folded) covering number of a host graph w.r.t. that template class

Download full-text


Available from: Torsten Ueckerdt
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: A new, constructive proof with a small explicit constant is given to the Erd\H{o}s-Pyber theorem which says that the edges of a graph on $n$ vertices can be partitioned into complete bipartite subgraphs so that every vertex is covered at most $O(n/\log n)$ times. The theorem is generalized to uniform hypergraphs. Similar bounds with smaller constant value is provided for fractional partitioning both for graphs and for uniform hypergraphs. We show that these latter constants cannot be improved by more than a factor of 1.89 even for fractional covering by arbitrary complete multipartite subgraphs or subhypergraphs. In the case every vertex of the graph is connected to at least $n-m$ other vertices, we prove the existence of a fractional covering of the edges by complete bipartite graphs such that every vertex is covered at most $O(m/\log m)$ times, with only a slightly worse explicit constant. This result also generalizes to uniform hypergraphs. Our results give new improved bounds on the complexity of graph and uniform hypergraph based secret sharing schemes, and show the limits of the method at the same time.
    Preview · Article · Nov 2013 · Graphs and Combinatorics
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: We investigate edge-intersection graphs of paths in the plane grid, regarding a parameter called the bend-number. I.e., every vertex is represented by a grid path and two vertices are adjacent if and only if the two grid paths share at least one grid-edge. The bend-number is the minimum $k$ such that grid-paths with at most $k$ bends each suffice to represent a given graph. This parameter is related to the interval-number and the track-number of a graph. We show that for every $k$ there is a graph with bend-number $k$. Moreover we provide new upper and lower bounds of the bend-number of graphs in terms of degeneracy, treewidth, edge clique covers and the maximum degree. Furthermore we give bounds on the bend-number of $K_{m,n}$ and determine it exactly for some pairs of $m$ and $n$. Finally, we prove that recognizing single-bend graphs is NP-complete, providing the first such result in this field.
    Full-text · Article · Apr 2014 · Discrete Applied Mathematics
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: Motivated by applications in graph drawing and information visualization, we examine the planar split thickness of a graph, that is, the smallest $k$ such that the graph is $k$-splittable into a planar graph. A $k$-split operation substitutes a vertex $v$ by at most $k$ new vertices such that each neighbor of $v$ is connected to at least one of the new vertices. We first examine the planar split thickness of complete and complete bipartite graphs. We then prove that it is NP-hard to recognize graphs that are $2$-splittable into a planar graph, and show that one can approximate the planar split thickness of a graph within a constant factor. If the treewidth is bounded, then we can even verify $k$-splittablity in linear time, for a constant $k$.
    Full-text · Article · Dec 2015
Show more