46002685
3-Restricted Edge Connectivity Of Vertex Transitive Graphs
204989114
On the numbers of spanning trees and Eulerian tours in generalized de Bruijn graphs
Abstract In this paper, we give the spectra and characteristic polynomials of generalized de Bruijn graphs. By Tutte's Theorem we can obtain the number of spanning trees and the number of Eulerian tours in such graphs.
5788058
Heredity of the index of convergence of the line digraph
Let D be a digraph, and let Ln(D) and k(D) denote the nth iterated line digraph of D and the index of convergence of D, respectively. We prove that if D contains neither sources nor sinks, then: (1) k(Ln(D))=k(D)=0 if every connected component of D is a directed cycle. (2) k(Ln(D))=k(D)+n if there exists at least one connected component of D that is not a directed cycle. Finally, we prove that if D has no sources or no sinks then k(D) ≤ k(L(D)) ≤ k(D)+1.
29766053
Construction for bicritical graphs and k-extendable bipartite graphs
A graph G is said to be bicritical if G-u-v has a perfect matching for every choice of a pair of points u and v. Bicritical graphs play a central role in decomposition theory of elementary graphs with respect to perfect matchings. As Plummer pointed out many times, the structure of bicritical graphs is far from completely understood. This paper presents a concise structure characterization on bicritical graphs in terms of factor-critical graphs and transversals of hypergraphs. A connected graph G with at least 2k+2 points is said to be k-extendable if it contains a matching of k lines and every such matching is contained in a perfect matching. A structure characterization for k-extendable bipartite graphs is given in a recursive way. Furthermore, this paper presents an O(mn) algorithm for determining the extendability of a bipartite graph G, the maximum integer k such that G is k-extendable, where n is the number of points and m is the number of lines in G.
45771940
Smoothing Tensor-Valued Images Using Anisotropic Geodesic Diffusion
This paper considers the feature space of DT-MRI as a differential manifold with an affine-invariant metric. We generalise Di Zenzo’s structure tensor to tensor-valued images for edge detection. To improve the quality of the edges, we develop a generalised Perona-Malik method for smoothing tensor images. We demonstrate our algorithm on both synthetic and real DT-MRI data.
6097933
The Clar covering polynomial of hexagonal systems III
Abstract The Clar covering polynomial is a recently proposed concept of hexagonal systems by which some important topological indices such as perfect matching count, Clar number, first Herndon number, etc., can be easily obtained. In this paper we establish a relationship between the Clar covering polynomial and sextet polynomial. A lower bound of the Clar number and some properties of coefficients of the Clar covering polynomial are thus deduced. It is mentioned that the summation of coefficients of Clar covering polynomial can be used to calculate the number of perfect matchings of a certain kind of polyominoes relating to crystal physics.
1746401
On the Effective Enumerability of NP Problems
In the field of computational optimization, it is often the case that we are given an instance of an NP problem and asked to enumerate the first few best solutions to the instance. Motivated by this, we propose in this paper a new framework to measure the effective enumerability of NP optimization problems. More specifically, given an instance of an NP problem, we consider the parameterized problem of enumerating a given number of best solutions to the instance, and study its average complexity in terms of the number of solutions. Our framework is different from the previously-proposed ones. For example, although it is known that counting the number of k-paths in a graph is #W[1]-complete, we present a fixed-parameter enumeration algorithm for the problem. We show that most algorithmic techniques for fixed-parameter tractable problems, such as search trees, color coding, and bounded treewidth, can be used for parameterized enumerations. In addition, we design elegant and new enumeration techniques and show how to generate small-size structures and enumerate solutions efficiently.
14205748
Edge Detection and Anisotropic Diffusion for Tensor-Valued Images
This paper considers feature space of DT-MRI as a differential manifold with an affine-invariant metric, where we generalise Di Zenzo's structure tensor to tensor-valued images for edge detection. To improve the quality of the edges, we smooth out the noise by generalised Perona-Malik method for tensor images. We demonstrate our algorithm on both synthetic and real DT-MRI data.