A path (resp. cycle) decomposition of a graph G is a set of edge-disjoint paths (resp. cycles) of G that covers the edge set of G. Gallai (1966) conjectured that every graph on n vertices admits a path decomposition of size at most ⌊(n+ 1)/2⌋, and Hajós (1968) conjectured that every Eulerian graph on n vertices admits a cycle decomposition of size at most ⌊(n−1)/2⌋. Gallai’s Conjecture was veri...

We present an overview of the design and test of an image processing procedure for detecting all important anatomical structures in color fundus images. These structures are the optic disk, the macula and the retinal network. The algorithm proceeds through five main steps: (1) automatic mask generation using pixels value statistics and color threshold, (2) visual image quality assessment using ...

We show that every sufficiently large r-regular digraph G which has linear degree and is a robust outexpander has an approximate decomposition into edge-disjoint Hamilton cycles, i.e. G contains a set of r−o(r) edge-disjoint Hamilton cycles. Here G is a robust outexpander if for every set S which is not too small and not too large, the ‘robust’ outneighbourhood of S is a little larger than S. T...

Let K be a general nite commutative ring. We refer to a family gn, n = 1, 2, . . . of bijective polynomial multivariate maps of K as a family with invertible decomposition gn = g ng 2 n . . . g k n, such that the knowledge of the composition of g n allows computation of g i n for O(n ) (s > 0) elementary steps. A polynomial map g is stable if all non-identical elements of kind g, t > 0 are of t...

In this paper we first prove that if the edge set of an undirected graph is the disjoint union of two of its spanning trees, then for every subset P of edges there exists a spanning tree decomposition that cuts P into two (almost) equal parts. The main result of the paper is a further extension of this claim: If the edge set of a graph is the disjoint union of two of its spanning trees, then fo...

The concept of H-decompositions of graphs was first introduced by Erdös, Goodman and Pósa in 1966, who were motivated by the problem of representing graphs by set intersections. Given graphs G and H, an H-decomposition of G is a partition of the edge set of G such that each part is either a single edge or forms a graph isomorphic to H. Let be the smallest number  , n H    , such that, any g...

We consider edge-decompositions of regular graphs into isomorphic paths. An m-PPD (perfect path decomposition) is a decomposition of a graph into paths of length m such that every vertex is an end of exactly two paths. An m-PPDC (perfect path double cover) is a covering of the edges by paths of length m such that every edge is covered exactly two times and every vertex is an end of exactly two ...

Denoising or usually known as noise reduction is one of the most essential processes for digital image processing. The goal of denoising is how to remove the noise while keeping the important image features as much as possible. In this paper, a denoising algorithm using adaptive block-based singular value decomposition filtering is presented. The proposed approach, instead of applying block-bas...

A path decomposition of a graph G is a collection of edge-disjoint paths of G that covers the edge set of G. Gallai (1968) conjectured that every connected graph on n vertices admits a path decomposition of cardinality at most ⌊(n + 1)/2⌋. Gallai’s Conjecture has been verified for many classes of graphs. In particular, Lovász (1968) verified this conjecture for graphs with at most one vertex wi...

Star graphs are Cayley graphs of symmetric groups of permutations, with transpositions as the generating sets. A star graph is a preferred interconnection network topology to a hypercube for its ability to connect a greater number of nodes with lower degree. However, an attractive property of the hypercube is that it has a Hamiltonian decomposition, i.e. its edges can be partitioned into disjoi...