On the use of graphs in discrete tomography

In this tutorial paper, we consider the basic image reconstruction problem which stems from discrete tomography. We derive a graph theoretical model and we explore some variations and extensions of this model. This allows us to establish connections with scheduling and timetabling applications. The complexity status of these problems is studied and we exhibit some polynomially solvable cases. We show how various classical techniques of operations research like matching, 2-SAT, network flows are applied to derive some of these result

Delineating the concept of self-management in chronic conditions: a concept analysis.

ObjectivesSelf-management is a concept frequently used within healthcare but lacks consensus. It is the aim of this study to clarify the concept.DesignConcept analysis according to Walker and Avant, comprises eight steps: select concept, determine purpose, identify uses, determine defining attributes, identify model case, identify additional cases, identify antecedents and consequences and define empirical referents. Sources used: PubMed, Scopus and Web of Science.resultsTen attributes delineating the concept have been identified and organised into three groups. Group (a): person-oriented attributes: the person must (1) actively take part in the care process, (2) take responsibility for the care process and (3) have a positive way of coping with adversity. Group (b): person-environment-oriented attributes: (4) the person must be informed about the condition, disease and treatment and self-management, (5) should be individualised, which entails expressing needs, values and priorities, (6) requires openness to ensure a reciprocal partnership with healthcare providers and (7) demands openness to social support. Finally, Group (c): summarising attributes: self-management (8) is a lifetime task, (9) assumes personal skills and (10) encompasses the medical, role and emotional management.ConclusionsThe findings of this study recognise the complexity of the concept, but also show the need for further investigation to make the concept more measurable. Clarity about the concept will enhance understanding and facilitate implementation in self-management programmes for chronic conditions

Label-free peptide profiling of Orbitrap™ full mass spectra

Background. We developed a new version of the open source software package Peptrix that can yet compare large numbers of Orbitrap™ LC-MS data. The peptide profiling results for Peptrix on MS1 spectra were compared with those obtained from a small selection of open source and commercial software packages: msInspect, Sieve™ and Progenesis™. The properties compared in these packages were speed, total number of detected masses, redundancy of masses, reproducibility in numbers and CV of intensity, overlap of masses, and differences in peptide peak intensities. Reproducibility measurements were taken for the different MS1 software applications by measuring in triplicate a complex peptide mixture of immunoglobulin on the Orbitrap™ mass spectrometer. Values of peptide masses detected from the high intensity peaks of the MS1 spectra by peptide profiling were verified with values of the MS2 fragmented and sequenced masses that resulted in protein identifications with a significant score. Findings. Peptrix finds about the same number of peptide features as the other packages, but peptide masses are in some cases approximately 5 to 10 times less redundant present in the peptide profile matrix. The Peptrix profile matrix displays the largest overlap when comparing the number of masses in a pair between two software applications. The overlap of peptide masses between software packages of low intensity peaks in the spectra is remarkably low with about 50% of the detected masses in the individual packages. Peptrix does not differ from the other packages in detecting 96% of the masses that relate to highly abundant sequenced proteins. MS1 peak intensities vary between the applications in a non linear way as they are not processed using the same method. Conclusions. Peptrix is capable of peptide profiling using Orbitrap™ files and finding differential expressed peptides in body fluid and tissue samples. The number of peptide masses detected in Orbitrap™ files can be increased by using more MS1 peptide profiling applications, including Peptrix, since it appears from the comparison of Peptrix with the other applications that all software packages have likely a high false negative rate of low intensity peptide peaks (missing peptides)

A tutorial on the use of graphs in discrete tomography

Starting from the basic image reconstruction problem in discrete tomography some graph theoretical models are proposed. This suggests the study of some variations and extensions of the basic problem. Applications in scheduling and timetabling are described. We exhibit polynomially solvable cases and we review the complexity status of some related problems

Bicolored Matchings in Some Classes of Graphs

We consider the problem of finding in a graph a set R of edges to be colored in red so that there are maximum matchings having some prescribed numbers of red edges. For regular bipartite graphs with n nodes on each side, we give sufficient conditions for the existence of a set R with |R| = n + 1 such that perfect matchings with k red edges exist for all k, 0 ≤ k ≤ n. Given two integers p < q we also determine the minimum cardinality of a set R of red edges such that there are perfect matchings with p red edges and with q red edges. For 3-regular bipartite graphs, we show that if p ≤ 4 there is a set R with |R| = p for which perfect matchingsMk exist with |Mk ∩R| ≤ k for all k ≤ p. For trees we design a linear time algorithm to determine a minimum set R of red edges such that there exist maximum matchings with k red edges for the largest possible number of values of k

Graph coloring with cardinality constraints on the neighborhoods

Extensions and variations of the basic problem of graph coloring are introduced. The problem consists essentially in finding in a graph a k-coloring, i.e., a partition (V_1,\cdots,V_k) of the vertex set of G such that, for some specified neighborhood \tilde|{N}(v) of each vertex v, the number of vertices in \tilde|{N}(v)\cap V_i is (at most) a given integer h_i^v. The complexity of some variations is discussed according to \tilde|{N}(v), which may be the usual neighbors, or the vertices at distance at most 2, or the closed neighborhood of v (v and its neighbors). Polynomially solvable cases are exhibited (in particular when is a special tree)

d-Transversals of stable sets and vertex covers in weighted bipartite graphs

Let G = (V , E) be a graph in which every vertex v ∈ V has a weight w(v)>=0 and a cost c(v) >=0. Let SG be the family of all maximum-weight stable sets in G. For any integer d 0, a minimum d-transversal in the graph G with respect to SG is a subset of vertices T ⊆ V of minimum total cost such that |T ∩ S| d for every S ∈ SG. In this paper, we present a polynomial-time algorithm to determine minimum d-transversals in bipartite graphs. Our algorithm is based on a characterization of maximum-weight stable sets in bipartite graphs. We also derive results on minimum d-transversals of minimum-weight vertex covers in weighted bipartite graphs

On a graph coloring problem arising from discrete tomography

An extension of the basic image reconstruction problem in discrete tomography is considered: given a graph G = (V,E) and a family equation image of chains Pi together with vectors h(Pi) = (h1, . . . , hik), one wants to find a partition V1,…,Vk of V such that for each Pi and each color j, |Vj ∩ Pi| = hij. An interpretation in terms of scheduling is presented. We consider special cases of graphs and identify polynomially solvable cases; general complexity results are established in this case and also in the case where V1,...Vk is required to be a proper vertex k-coloring of G. Finally, we examine also the case of (proper) edge k-colorings and determine its complexity status

Degree-constrained edge partitioning in graphs arising from discrete tomography

Starting from the basic problem of reconstructing a 2-dimensional im- age given by its projections on two axes, one associates a model of edge coloring in a complete bipartite graph. The complexity of the case with k = 3 colors is open. Variations and special cases are considered for the case k = 3 colors where the graph corresponding to the union of some color classes (for instance colors 1 and 2) has a given structure (tree, vertex- disjoint chains, 2-factor, etc.). We also study special cases corresponding to the search of 2 edge-disjoint chains or cycles going through speci ed vertices. A variation where the graph is oriented is also presented. In addition we explore similar problems for the case where the under- lying graph is a complete graph (instead of a complete bipartite graph)