3,953 research outputs found
Some NP-complete edge packing and partitioning problems in planar graphs
Graph packing and partitioning problems have been studied in many contexts,
including from the algorithmic complexity perspective. Consider the packing
problem of determining whether a graph contains a spanning tree and a cycle
that do not share edges. Bern\'ath and Kir\'aly proved that this decision
problem is NP-complete and asked if the same result holds when restricting to
planar graphs. Similarly, they showed that the packing problem with a spanning
tree and a path between two distinguished vertices is NP-complete. They also
established the NP-completeness of the partitioning problem of determining
whether the edge set of a graph can be partitioned into a spanning tree and a
(not-necessarily spanning) tree. We prove that all three problems remain
NP-complete even when restricted to planar graphs.Comment: 6 pages, 2 figure
The Galois Complexity of Graph Drawing: Why Numerical Solutions are Ubiquitous for Force-Directed, Spectral, and Circle Packing Drawings
Many well-known graph drawing techniques, including force directed drawings,
spectral graph layouts, multidimensional scaling, and circle packings, have
algebraic formulations. However, practical methods for producing such drawings
ubiquitously use iterative numerical approximations rather than constructing
and then solving algebraic expressions representing their exact solutions. To
explain this phenomenon, we use Galois theory to show that many variants of
these problems have solutions that cannot be expressed by nested radicals or
nested roots of low-degree polynomials. Hence, such solutions cannot be
computed exactly even in extended computational models that include such
operations.Comment: Graph Drawing 201
Strongly Monotone Drawings of Planar Graphs
A straight-line drawing of a graph is a monotone drawing if for each pair of
vertices there is a path which is monotonically increasing in some direction,
and it is called a strongly monotone drawing if the direction of monotonicity
is given by the direction of the line segment connecting the two vertices.
We present algorithms to compute crossing-free strongly monotone drawings for
some classes of planar graphs; namely, 3-connected planar graphs, outerplanar
graphs, and 2-trees. The drawings of 3-connected planar graphs are based on
primal-dual circle packings. Our drawings of outerplanar graphs are based on a
new algorithm that constructs strongly monotone drawings of trees which are
also convex. For irreducible trees, these drawings are strictly convex
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)
Nested hierarchies in planar graphs
We construct a partial order relation which acts on the set of 3-cliques of a
maximal planar graph G and defines a unique hierarchy. We demonstrate that G is
the union of a set of special subgraphs, named `bubbles', that are themselves
maximal planar graphs. The graph G is retrieved by connecting these bubbles in
a tree structure where neighboring bubbles are joined together by a 3-clique.
Bubbles naturally provide the subdivision of G into communities and the tree
structure defines the hierarchical relations between these communities
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