169 research outputs found
Paired 2-disjoint path covers of burnt pancake graphs with faulty elements
The burnt pancake graph is the Cayley graph of the hyperoctahedral
group using prefix reversals as generators. Let and be any
two pairs of distinct vertices of for . We show that there are
and paths whose vertices partition the vertex set of even if
has up to faulty elements. On the other hand, for every
there is a set of faulty edges or faulty vertices for which such a
fault-free disjoint path cover does not exist.Comment: 14 pages, 4 figure
Optimal Vertex Fault Tolerant Spanners (for fixed stretch)
A -spanner of a graph is a sparse subgraph whose shortest path
distances match those of up to a multiplicative error . In this paper we
study spanners that are resistant to faults. A subgraph is an
vertex fault tolerant (VFT) -spanner if is a -spanner
of for any small set of vertices that might "fail." One
of the main questions in the area is: what is the minimum size of an fault
tolerant -spanner that holds for all node graphs (as a function of ,
and )? This question was first studied in the context of geometric
graphs [Levcopoulos et al. STOC '98, Czumaj and Zhao SoCG '03] and has more
recently been considered in general undirected graphs [Chechik et al. STOC '09,
Dinitz and Krauthgamer PODC '11].
In this paper, we settle the question of the optimal size of a VFT spanner,
in the setting where the stretch factor is fixed. Specifically, we prove
that every (undirected, possibly weighted) -node graph has a
-spanner resilient to vertex faults with edges, and this is fully optimal (unless the famous Erdos Girth
Conjecture is false). Our lower bound even generalizes to imply that no data
structure capable of approximating similarly can
beat the space usage of our spanner in the worst case. We also consider the
edge fault tolerant (EFT) model, defined analogously with edge failures rather
than vertex failures. We show that the same spanner upper bound applies in this
setting. Our data structure lower bound extends to the case (and hence we
close the EFT problem for -approximations), but it falls to for . We leave it as an open problem to
close this gap.Comment: To appear in SODA 201
Packing and embedding large subgraphs
This thesis contains several embedding results for graphs in both random and non random settings.
Most notably, we resolve a long standing conjecture that the threshold probability for Hamiltonicity in the random binomial subgraph of the hypercube equals . %posed e.g.~by Bollob\'as,
In Chapter 2 we obtain the following perturbation result regarding the hypercube \cQ^n:
if H\subseteq\cQ^n satisfies with fixed and we consider a random binomial subgraph \cQ^n_p of \cQ^n with fixed, then with high probability H\cup\cQ^n_p contains edge-disjoint Hamilton cycles, for any fixed .
This result is part of a larger volume of work where we also prove the corresponding hitting time result for Hamiltonicity.
In Chapter 3 we move to a non random setting. %to a deterministic one.
%Instead of embedding a single Hamilton cycle our result concerns packing more general families of graphs into a fixed host graph.
Rather than pack a small number of Hamilton cycles into a fixed host graph, our aim is to achieve optimally sized packings of more general families of graphs.
More specifically, we provide a degree condition on a regular -vertex graph which ensures the existence of a near optimal packing of any family of bounded degree -vertex -chromatic separable graphs into .
%In general, this degree condition is best possible.
%In particular, this yields an approximate version of the tree packing conjecture
%in the setting of regular host graphs of high degree.
%Similarly, our result implies approximate versions of the Oberwolfach problem,
%the Alspach problem and the existence of resolvable designs in the setting of
%regular host graphs of high degree.
In particular, this yields approximate versions of the the tree packing conjecture, the Oberwolfach problem,
the Alspach problem and the existence of resolvable designs in the setting of regular host graphs of high degree
Broadcasting in highly connected graphs
Throughout history, spreading information has been an important task. With computer networks expanding, fast and reliable dissemination of messages became a problem of interest for computer scientists. Broadcasting is one category of information dissemination that transmits a message from a single originator to all members of the network. In the past five decades the problem has been studied by many researchers and all have come to demonstrate that despite its easy definition, the problem of broadcasting does not have trivial properties and symmetries. For general graphs, and even for some very restricted classes of graphs, the question of finding the broadcast time and scheme remains NP-hard. This work uses graph theoretical concepts to explore mathematical bounds on how fast information can be broadcast in a network. The connectivity of a graph is a measure to assess how separable the graph is, or in other words how many machines in a network will have to fail to disrupt communication between all machines in the network.
We initiate the study of finding upper bounds on broadcast time b(G) in highly connected graphs. In particular, we give upper bounds on b(G) for k-connected graphs and graphs with a large minimum degree.
We explore 2-connected (biconnected) graphs and broadcasting in them. Using Whitney's open ear decomposition in an inductive proof we propose broadcast schemes that achieve an upper bound of ceil(n/2) for classical broadcasting as well as similar bounds for multiple originators. Exploring further, we use a matching-based approach to prove an upper bound of ceil(log(k)) + ceil(n/k) - 1 for all k-connected graphs. For many infinite families of graphs, these bounds are tight.
Discussion of broadcasting in highly connected graphs leads to an exploration of dependence between the minimum degree in the graph and the broadcast time of the latter. By using similar techniques and arguments we show that if all vertices of the graph are neighboring linear numbers of vertices, then information dissemination in the graph can be achieved in ceil(log(n)) + C time.
To the best of our knowledge, the bounds presented in our work are a novelty. Methods and questions proposed in this thesis open new pathways for research in broadcasting
Magnetic Material Modelling of Electrical Machines
The need for electromechanical energy conversion that takes place in electric motors, generators, and actuators is an important aspect associated with current development. The efficiency and effectiveness of the conversion process depends on both the design of the devices and the materials used in those devices. In this context, this book addresses important aspects of electrical machines, namely their materials, design, and optimization. It is essential for the design process of electrical machines to be carried out through extensive numerical field computations. Thus, the reprint also focuses on the accuracy of these computations, as well as the quality of the material models that are adopted. Another aspect of interest is the modeling of properties such as hysteresis, alternating and rotating losses and demagnetization. In addition, the characterization of materials and their dependence on mechanical quantities such as stresses and temperature are also considered. The reprint also addresses another aspect that needs to be considered for the development of the optimal global system in some applications, which is the case of drives that are associated with electrical machines
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