848 research outputs found
On the tractability of some natural packing, covering and partitioning problems
In this paper we fix 7 types of undirected graphs: paths, paths with
prescribed endvertices, circuits, forests, spanning trees, (not necessarily
spanning) trees and cuts. Given an undirected graph and two "object
types" and chosen from the alternatives above, we
consider the following questions. \textbf{Packing problem:} can we find an
object of type and one of type in the edge set of
, so that they are edge-disjoint? \textbf{Partitioning problem:} can we
partition into an object of type and one of type ?
\textbf{Covering problem:} can we cover with an object of type
, and an object of type ? This framework includes 44
natural graph theoretic questions. Some of these problems were well-known
before, for example covering the edge-set of a graph with two spanning trees,
or finding an - path and an - path that are
edge-disjoint. However, many others were not, for example can we find an
- path and a spanning tree that are
edge-disjoint? Most of these previously unknown problems turned out to be
NP-complete, many of them even in planar graphs. This paper determines the
status of these 44 problems. For the NP-complete problems we also investigate
the planar version, for the polynomial problems we consider the matroidal
generalization (wherever this makes sense)
Three ways to cover a graph
We consider the problem of covering an input graph with graphs from a
fixed covering class . The classical covering number of with respect to
is the minimum number of graphs from needed to cover the edges of
without covering non-edges of . We introduce a unifying notion of three
covering parameters with respect to , two of which are novel concepts only
considered in special cases before: the local and the folded covering number.
Each parameter measures "how far'' is from in a different way. Whereas
the folded covering number has been investigated thoroughly for some covering
classes, e.g., interval graphs and planar graphs, the local covering number has
received little attention.
We provide new bounds on each covering number with respect to the following
covering classes: linear forests, star forests, caterpillar forests, and
interval graphs. The classical graph parameters that result this way are
interval number, track number, linear arboricity, star arboricity, and
caterpillar arboricity. As input graphs we consider graphs of bounded
degeneracy, bounded degree, bounded tree-width or bounded simple tree-width, as
well as outerplanar, planar bipartite, and planar graphs. For several pairs of
an input class and a covering class we determine exactly the maximum ordinary,
local, and folded covering number of an input graph with respect to that
covering class.Comment: 20 pages, 4 figure
Partitions and Coverings of Trees by Bounded-Degree Subtrees
This paper addresses the following questions for a given tree and integer
: (1) What is the minimum number of degree- subtrees that partition
? (2) What is the minimum number of degree- subtrees that cover
? We answer the first question by providing an explicit formula for the
minimum number of subtrees, and we describe a linear time algorithm that finds
the corresponding partition. For the second question, we present a polynomial
time algorithm that computes a minimum covering. We then establish a tight
bound on the number of subtrees in coverings of trees with given maximum degree
and pathwidth. Our results show that pathwidth is the right parameter to
consider when studying coverings of trees by degree-3 subtrees. We briefly
consider coverings of general graphs by connected subgraphs of bounded degree
Boxicity and separation dimension
A family of permutations of the vertices of a hypergraph is
called 'pairwise suitable' for if, for every pair of disjoint edges in ,
there exists a permutation in in which all the vertices in one
edge precede those in the other. The cardinality of a smallest such family of
permutations for is called the 'separation dimension' of and is denoted
by . Equivalently, is the smallest natural number so that
the vertices of can be embedded in such that any two
disjoint edges of can be separated by a hyperplane normal to one of the
axes. We show that the separation dimension of a hypergraph is equal to the
'boxicity' of the line graph of . This connection helps us in borrowing
results and techniques from the extensive literature on boxicity to study the
concept of separation dimension.Comment: This is the full version of a paper by the same name submitted to
WG-2014. Some results proved in this paper are also present in
arXiv:1212.6756. arXiv admin note: substantial text overlap with
arXiv:1212.675
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