156 research outputs found
On the size of maximally non-hamiltonian digraphs
A graph is called maximally non-hamiltonian if it is non-hamiltonian, yet for any two non-adjacent vertices there exists a hamiltonian path between them. In this paper, we naturally extend the concept to directed graphs and bound their size from below and above. Our results on the lower bound constitute our main contribution, while the upper bound can be obtained using a result of Lewin, but we give here a different proof. We describe digraphs attaining the upper bound, but whether our lower bound can be improved remains open
Further topics in connectivity
Continuing the study of connectivity, initiated in §4.1 of the Handbook, we survey here some (sufficient) conditions under which a graph or digraph has a given connectivity or edge-connectivity. First, we describe results concerning maximal (vertex- or edge-) connectivity. Next, we deal with conditions for having (usually lower) bounds for the connectivity parameters. Finally, some other general connectivity measures, such as one instance of the so-called “conditional connectivity,” are considered.
For unexplained terminology concerning connectivity, see §4.1.Peer ReviewedPostprint (published version
Regular configurations and TBR graphs
PhD 2009 QMThis thesis consists of two parts: The first one is concerned with the
theory and applications of regular configurations; the second one is devoted
to TBR graphs.
In the first part, a new approach is proposed to study regular configurations,
an extremal arrangement of necklaces formed by a given number
of red beads and black beads. We first show that this concept is closely related
to several other concepts studied in the literature, such as balanced
words, maximally even sets, and the ground states in the Kawasaki-Ising
model. Then we apply regular configurations to solve the (vertex) cycle
packing problem for shift digraphs, a family of Cayley digraphs.
TBR is one of widely used tree rearrangement operationes, and plays
an important role in heuristic algorithms for phylogenetic tree reconstruction.
In the second part of this thesis we study various properties
of TBR graphs, a family of graphs associated with the TBR operation.
To investigate the degree distribution of the TBR graphs, we also study
-index, a concept introduced to measure the shape of trees. As an interesting
by-product, we obtain a structural characterization of good trees,
a well-known family of trees that generalizes the complete binary trees
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)
- …