3,399 research outputs found
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)
Superization and (q,t)-specialization in combinatorial Hopf algebras
We extend a classical construction on symmetric functions, the superization
process, to several combinatorial Hopf algebras, and obtain analogs of the
hook-content formula for the (q,t)-specializations of various bases. Exploiting
the dendriform structures yields in particular (q,t)-analogs of the
Bjorner-Wachs q-hook-length formulas for binary trees, and similar formulas for
plane trees.Comment: 30 page
The Planar Tree Packing Theorem
Packing graphs is a combinatorial problem where several given graphs are
being mapped into a common host graph such that every edge is used at most
once. In the planar tree packing problem we are given two trees T1 and T2 on n
vertices and have to find a planar graph on n vertices that is the
edge-disjoint union of T1 and T2. A clear exception that must be made is the
star which cannot be packed together with any other tree. But according to a
conjecture of Garc\'ia et al. from 1997 this is the only exception, and all
other pairs of trees admit a planar packing. Previous results addressed various
special cases, such as a tree and a spider tree, a tree and a caterpillar, two
trees of diameter four, two isomorphic trees, and trees of maximum degree
three. Here we settle the conjecture in the affirmative and prove its general
form, thus making it the planar tree packing theorem. The proof is constructive
and provides a polynomial time algorithm to obtain a packing for two given
nonstar trees.Comment: Full version of our SoCG 2016 pape
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