121,747 research outputs found
Separating path systems
We study separating systems of the edges of a graph where each member of the
separating system is a path. We conjecture that every -vertex graph admits a
separating path system of size and prove this in certain interesting
special cases. In particular, we establish this conjecture for random graphs
and graphs with linear minimum degree. We also obtain tight bounds on the size
of a minimal separating path system in the case of trees.Comment: 21 pages, fixed misprints, Journal of Combinatoric
Locating-dominating sets and identifying codes in graphs of girth at least 5
Locating-dominating sets and identifying codes are two closely related
notions in the area of separating systems. Roughly speaking, they consist in a
dominating set of a graph such that every vertex is uniquely identified by its
neighbourhood within the dominating set. In this paper, we study the size of a
smallest locating-dominating set or identifying code for graphs of girth at
least 5 and of given minimum degree. We use the technique of vertex-disjoint
paths to provide upper bounds on the minimum size of such sets, and construct
graphs who come close to meet these bounds.Comment: 20 pages, 9 figure
Coordination of Dynamic Software Components with JavaBIP
JavaBIP allows the coordination of software components by clearly separating
the functional and coordination aspects of the system behavior. JavaBIP
implements the principles of the BIP component framework rooted in rigorous
operational semantics. Recent work both on BIP and JavaBIP allows the
coordination of static components defined prior to system deployment, i.e., the
architecture of the coordinated system is fixed in terms of its component
instances. Nevertheless, modern systems, often make use of components that can
register and deregister dynamically during system execution. In this paper, we
present an extension of JavaBIP that can handle this type of dynamicity. We use
first-order interaction logic to define synchronization constraints based on
component types. Additionally, we use directed graphs with edge coloring to
model dependencies among components that determine the validity of an online
system. We present the software architecture of our implementation, provide and
discuss performance evaluation results.Comment: Technical report that accompanies the paper accepted at the 14th
International Conference on Formal Aspects of Component Softwar
Optimal transportation, topology and uniqueness
The Monge-Kantorovich transportation problem involves optimizing with respect
to a given a cost function. Uniqueness is a fundamental open question about
which little is known when the cost function is smooth and the landscapes
containing the goods to be transported possess (non-trivial) topology. This
question turns out to be closely linked to a delicate problem (# 111) of
Birkhoff [14]: give a necessary and sufficient condition on the support of a
joint probability to guarantee extremality among all measures which share its
marginals. Fifty years of progress on Birkhoff's question culminate in Hestir
and Williams' necessary condition which is nearly sufficient for extremality;
we relax their subtle measurability hypotheses separating necessity from
sufficiency slightly, yet demonstrate by example that to be sufficient
certainly requires some measurability. Their condition amounts to the vanishing
of the measure \gamma outside a countable alternating sequence of graphs and
antigraphs in which no two graphs (or two antigraphs) have domains that
overlap, and where the domain of each graph / antigraph in the sequence
contains the range of the succeeding antigraph (respectively, graph). Such
sequences are called numbered limb systems. We then explain how this
characterization can be used to resolve the uniqueness of Kantorovich solutions
for optimal transportation on a manifold with the topology of the sphere.Comment: 36 pages, 6 figure
Covering and Separation for Permutations and Graphs
This is a thesis of two parts, focusing on covering and separation topics of extremal combinatorics and graph theory, two major themes in this area. They entail the existence and properties of collections of combinatorial objects which together either represent all objects (covering) or can be used to distinguish all objects from each other (separation). We will consider a range of problems which come under these areas. The first part will focus on shattering k-sets with permutations. A family of permutations is said to shatter a given k-set if the permutations cover all possible orderings of the k elements. In particular, we investigate the size of permutation families which cover t orders for every possible k-set as well as study the problem of determining the largest number of k-sets that can be shattered by a family with given size. We provide a construction for a small permutation family which shatters every k-set. We also consider constructions of large families which do not shatter any triple. The second part will be concerned with the problem of separating path systems. A separating path system for a graph is a family of paths where, for any two edges, there is a path containing one edge but not the other. The aim is to find the size of the smallest such family. We will study the size of the smallest separating path system for a range of graphs, including complete graphs, complete bipartite graphs, and lattice-type graphs. A key technique we introduce is the use of generator paths - constructed to utilise the symmetric nature of Kn. We continue this symmetric approach for bipartite graphs and study the limitations of the method. We consider lattice-type graphs as an example of the most efficient possible separating systems for any graph
Classification of Minimal Separating Sets in Low Genus Surfaces
Consider a surface and let . If is not
connected, then we say \emph{separates} , and we refer to as a
\emph{separating set} of . If separates , and no proper subset of
separates , then we say is a \emph{minimal separating set} of . In
this paper we use methods of computational combinatorial topology to classify
the minimal separating sets of the orientable surfaces of genus and
. The classification for genus 0 and 1 was done in earlier work, using
methods of algebraic topology.Comment: 24 pages, 5 figures, 2 tables (11 pages
Characterizing extremal digraphs for identifying codes and extremal cases of Bondy's theorem on induced subsets
An identifying code of a (di)graph is a dominating subset of the
vertices of such that all distinct vertices of have distinct
(in)neighbourhoods within . In this paper, we classify all finite digraphs
which only admit their whole vertex set in any identifying code. We also
classify all such infinite oriented graphs. Furthermore, by relating this
concept to a well known theorem of A. Bondy on set systems we classify the
extremal cases for this theorem
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