220 research outputs found
Connectedness of graphs and its application to connected matroids through covering-based rough sets
Graph theoretical ideas are highly utilized by computer science fields
especially data mining. In this field, a data structure can be designed in the
form of tree. Covering is a widely used form of data representation in data
mining and covering-based rough sets provide a systematic approach to this type
of representation. In this paper, we study the connectedness of graphs through
covering-based rough sets and apply it to connected matroids. First, we present
an approach to inducing a covering by a graph, and then study the connectedness
of the graph from the viewpoint of the covering approximation operators.
Second, we construct a graph from a matroid, and find the matroid and the graph
have the same connectedness, which makes us to use covering-based rough sets to
study connected matroids. In summary, this paper provides a new approach to
studying graph theory and matroid theory
Some characteristics of matroids through rough sets
At present, practical application and theoretical discussion of rough sets
are two hot problems in computer science. The core concepts of rough set theory
are upper and lower approximation operators based on equivalence relations.
Matroid, as a branch of mathematics, is a structure that generalizes linear
independence in vector spaces. Further, matroid theory borrows extensively from
the terminology of linear algebra and graph theory. We can combine rough set
theory with matroid theory through using rough sets to study some
characteristics of matroids. In this paper, we apply rough sets to matroids
through defining a family of sets which are constructed from the upper
approximation operator with respect to an equivalence relation. First, we prove
the family of sets satisfies the support set axioms of matroids, and then we
obtain a matroid. We say the matroids induced by the equivalence relation and a
type of matroid, namely support matroid, is induced. Second, through rough
sets, some characteristics of matroids such as independent sets, support sets,
bases, hyperplanes and closed sets are investigated.Comment: 13 page
Rough sets and matroidal contraction
Rough sets are efficient for data pre-processing in data mining. As a
generalization of the linear independence in vector spaces, matroids provide
well-established platforms for greedy algorithms. In this paper, we apply rough
sets to matroids and study the contraction of the dual of the corresponding
matroid. First, for an equivalence relation on a universe, a matroidal
structure of the rough set is established through the lower approximation
operator. Second, the dual of the matroid and its properties such as
independent sets, bases and rank function are investigated. Finally, the
relationships between the contraction of the dual matroid to the complement of
a single point set and the contraction of the dual matroid to the complement of
the equivalence class of this point are studied.Comment: 11 page
A central partition of molecular conformational space. II. Embedding 3D structures
A combinatorial model of molecular conformational space that was previously
developped (J. Gabarro-Arpa, Comp. Biol. and Chem. 27, (2003) 153-159), had the
drawback that structures could not be properly embedded beacause it lacked
explicit rotational symmetry. The problem can be circumvented by sorting the
elementary 3D components of a molecular system into a finite set of classes
that can be separately embedded. This also opens up the possibility of encoding
the dynamical states into a graph structure
Discrete Mathematics and Symmetry
Some of the most beautiful studies in Mathematics are related to Symmetry and Geometry. For this reason, we select here some contributions about such aspects and Discrete Geometry. As we know, Symmetry in a system means invariance of its elements under conditions of transformations. When we consider network structures, symmetry means invariance of adjacency of nodes under the permutations of node set. The graph isomorphism is an equivalence relation on the set of graphs. Therefore, it partitions the class of all graphs into equivalence classes. The underlying idea of isomorphism is that some objects have the same structure if we omit the individual character of their components. A set of graphs isomorphic to each other is denominated as an isomorphism class of graphs. The automorphism of a graph will be an isomorphism from G onto itself. The family of all automorphisms of a graph G is a permutation group
Graph Theory
Highlights of this workshop on structural graph theory included new developments on graph and matroid minors, continuous structures arising as limits of finite graphs, and new approaches to higher graph connectivity via tree structures
Graph Theory
Graph theory is a rapidly developing area of mathematics. Recent years have seen the development of deep theories, and the increasing importance of methods from other parts of mathematics. The workshop on Graph Theory brought together together a broad range of researchers to discuss some of the major new developments. There were three central themes, each of which has seen striking recent progress: the structure of graphs with forbidden subgraphs; graph minor theory; and applications of the entropy compression method. The workshop featured major talks on current work in these areas, as well as presentations of recent breakthroughs and connections to other areas. There was a particularly exciting selection of longer talks, including presentations on the structure of graphs with forbidden induced subgraphs, embedding simply connected 2-complexes in 3-space, and an announcement of the solution of the well-known Oberwolfach Problem
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