86 research outputs found
Editing to a Graph of Given Degrees
We consider the Editing to a Graph of Given Degrees problem that asks for a
graph G, non-negative integers d,k and a function \delta:V(G)->{1,...,d},
whether it is possible to obtain a graph G' from G such that the degree of v is
\delta(v) for any vertex v by at most k vertex or edge deletions or edge
additions. We construct an FPT-algorithm for Editing to a Graph of Given
Degrees parameterized by d+k. We complement this result by showing that the
problem has no polynomial kernel unless NP\subseteq coNP/poly
On the swap-distances of different realizations of a graphical degree sequence
One of the first graph theoretical problems which got serious attention
(already in the fifties of the last century) was to decide whether a given
integer sequence is equal to the degree sequence of a simple graph (or it is
{\em graphical} for short). One method to solve this problem is the greedy
algorithm of Havel and Hakimi, which is based on the {\em swap} operation.
Another, closely related question is to find a sequence of swap operations to
transform one graphical realization into another one of the same degree
sequence. This latter problem got particular emphases in connection of fast
mixing Markov chain approaches to sample uniformly all possible realizations of
a given degree sequence. (This becomes a matter of interest in connection of --
among others -- the study of large social networks.) Earlier there were only
crude upper bounds on the shortest possible length of such swap sequences
between two realizations. In this paper we develop formulae (Gallai-type
identities) for these {\em swap-distance}s of any two realizations of simple
undirected or directed degree sequences. These identities improves considerably
the known upper bounds on the swap-distances.Comment: to be publishe
The mixing time of the switch Markov chains: a unified approach
Since 1997 a considerable effort has been spent to study the mixing time of
switch Markov chains on the realizations of graphic degree sequences of simple
graphs. Several results were proved on rapidly mixing Markov chains on
unconstrained, bipartite, and directed sequences, using different mechanisms.
The aim of this paper is to unify these approaches. We will illustrate the
strength of the unified method by showing that on any -stable family of
unconstrained/bipartite/directed degree sequences the switch Markov chain is
rapidly mixing. This is a common generalization of every known result that
shows the rapid mixing nature of the switch Markov chain on a region of degree
sequences. Two applications of this general result will be presented. One is an
almost uniform sampler for power-law degree sequences with exponent
. The other one shows that the switch Markov chain on the
degree sequence of an Erd\H{o}s-R\'enyi random graph is asymptotically
almost surely rapidly mixing if is bounded away from 0 and 1 by at least
.Comment: Clarification
On Generalizations of Supereulerian Graphs
A graph is supereulerian if it has a spanning closed trail. Pulleyblank in 1979 showed that determining whether a graph is supereulerian, even when restricted to planar graphs, is NP-complete. Let and be the edge-connectivity and the minimum degree of a graph , respectively. For integers and , a graph is -supereulerian if for any disjoint edge sets with and , has a spanning closed trail that contains and avoids . This dissertation is devoted to providing some results on -supereulerian graphs and supereulerian hypergraphs.
In Chapter 2, we determine the value of the smallest integer such that every -edge-connected graph is -supereulerian as follows:
j(s,t) = \left\{ \begin{array}{ll} \max\{4, t + 2\} & \mbox{ if $0 \le s \le 1$, or $(s,t) \in \{(2,0), (2,1), (3,0),(4,0)\}$,} \\ 5 & \mbox{ if $(s,t) \in \{(2,2), (3,1)\}$,} \\ s + t + \frac{1 - (-1)^s}{2} & \mbox{ if $s \ge 2$ and $s+t \ge 5$. } \end{array} \right.
As applications, we characterize -supereulerian graphs when in terms of edge-connectivities, and show that when , -supereulerianicity is polynomially determinable.
In Chapter 3, for a subset with , a necessary and sufficient condition for to be a contractible configuration for supereulerianicity is obtained. We also characterize the -supereulerianicity of when . These results are applied to show that if is -supereulerian with , then for any permutation on the vertex set , the permutation graph is -supereulerian if and only if .
For a non-negative integer , a graph is -Hamiltonian if the removal of any vertices results in a Hamiltonian graph. Let and denote the smallest integer such that the iterated line graph is -supereulerian and -Hamiltonian, respectively. In Chapter 4, for a simple graph , we establish upper bounds for and . Specifically, the upper bound for the -Hamiltonian index sharpens the result obtained by Zhang et al. in [Discrete Math., 308 (2008) 4779-4785].
Harary and Nash-Williams in 1968 proved that the line graph of a graph is Hamiltonian if and only if has a dominating closed trail, Jaeger in 1979 showed that every 4-edge-connected graph is supereulerian, and Catlin in 1988 proved that every graph with two edge-disjoint spanning trees is a contractible configuration for supereulerianicity. In Chapter 5, utilizing the notion of partition-connectedness of hypergraphs introduced by Frank, Kir\\u27aly and Kriesell in 2003, we generalize the above-mentioned results of Harary and Nash-Williams, of Jaeger and of Catlin to hypergraphs by characterizing hypergraphs whose line graphs are Hamiltonian, and showing that every 2-partition-connected hypergraph is a contractible configuration for supereulerianicity.
Applying the adjacency matrix of a hypergraph defined by Rodr\\u27iguez in 2002, let be the second largest adjacency eigenvalue of . In Chapter 6, we prove that for an integer and a -uniform hypergraph of order with even, the minimum degree and , if , then is -edge-connected. %.
Some discussions are displayed in the last chapter. We extend the well-known Thomassen Conjecture that every 4-connected line graph is Hamiltonian to hypergraphs. The -supereulerianicity of hypergraphs is another interesting topic to be investigated in the future
Basic Neutrosophic Algebraic Structures and their Application to Fuzzy and Neutrosophic Models
The involvement of uncertainty of varying degrees when the total of the
membership degree exceeds one or less than one, then the newer mathematical
paradigm shift, Fuzzy Theory proves appropriate. For the past two or more
decades, Fuzzy Theory has become the potent tool to study and analyze
uncertainty involved in all problems. But, many real-world problems also abound
with the concept of indeterminacy. In this book, the new, powerful tool of
neutrosophy that deals with indeterminacy is utilized. Innovative neutrosophic
models are described. The theory of neutrosophic graphs is introduced and
applied to fuzzy and neutrosophic models. This book is organized into four
chapters. In Chapter One we introduce some of the basic neutrosophic algebraic
structures essential for the further development of the other chapters. Chapter
Two recalls basic graph theory definitions and results which has interested us
and for which we give the neutrosophic analogues. In this chapter we give the
application of graphs in fuzzy models. An entire section is devoted for this
purpose. Chapter Three introduces many new neutrosophic concepts in graphs and
applies it to the case of neutrosophic cognitive maps and neutrosophic
relational maps. The last section of this chapter clearly illustrates how the
neutrosophic graphs are utilized in the neutrosophic models. The final chapter
gives some problems about neutrosophic graphs which will make one understand
this new subject.Comment: 149 pages, 130 figure
Topology of complex networks: models and analysis
There is a large variety of real-world phenomena that can be modelled and analysed as networks. Part of this variety is reflected in the diversity of network classes that are used to model these phenomena. However, the differences between network classes are not always taken into account in their analysis. This thesis carefully addresses how to deal with distinct classes of networks in two different contexts. First, the well-known switching model has been used to randomise different classes of networks, and is typically referred to as the switching model. We argue that really we should be talking about a family of switching models. Ignoring the distinction between the switching model with respect to different network classes has lead to biased sampling. Given that the most common use of the switching model is as a null-model, it is critical that it samples without bias. We provide a comprehensive analysis of the switching model with respect to nine classes of networks and prove under which conditions sampling is unbiased for each class. Recently the Curveball algorithm was introduced as a faster approach to network randomisation. We prove that the Curveball algorithm samples without bias; a position that was previously implied, but unproven. Furthermore, we show that the Curveball algorithm provides a flexible framework for network randomisation by introducing five variations with respect to different network classes. We compare the switching models and Curveball algorithms to several other random network models. As a result of our findings, we recommend using the configuration model for multi-graphs with self-loops, the Curveball algorithm for networks without multiple edges or without self-loops and the ordered switching model for directed acyclic networks. Second, we extend the theory of motif analysis to directed acyclic networks. We establish experimentally that there is no difference in the motifs detected by existing motif analysis methods and our customised method. However, we show that there are differences in the detected anti-motifs. Hence, we recommend taking into account the acyclic nature of directed acyclic networks. Network science is a young and active field of research. Most existing network measures originate in statistical mechanics and focus on statistics of local network properties. Such statistics have proven very useful. However, they do not capture the complete structure of a network. In this thesis we present experimental results on two novel network analysis techniques. First, at the local level, we show that the neighbourhood of a node is highly distinctive and has the potential to match unidentified entities across networks. Our motivation is the identification of individuals across dark social networks hidden in recorded networks. Second, we present results of the application of persistent homology to network analysis. This recently introduced technique from topological data analysis offers a new perspective on networks: it describes the mesoscopic structure of a network. Finally, we used persistent homology for a classification problem in pharmaceutical science. This is a novel application of persistent homology. Our analysis shows that this is a promising approach for the classification of lipid formulations
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