Drago Bokal

A generalization of Hungarian method and Hall's theorem with applications in wireless sensor networks

In this paper, we consider various problems concerning quasi-matchings and semi-matchings in bipartite graphs, which generalize the classical problem of determining a perfect matching in bipartite graphs. We prove a generalization of Halls marriage theorem, and present an algorithm that solves the problem of determining a lexicographically minimum g -quasi-matching (that is a set F of edges in a bipartite graph such that in one set of the bipartition every vertex v has at least g ( v ) incident edges from F , where g is a so-called need mapping, while on the other side of the bipartition the distribution of degrees with respect to F is lexicographically minimum). We obtain that finding a lexicographically minimum quasi-matching is equivalent to minimizing any strictly convex function on the degrees of the A-side of a quasi-matching and use this fact to prove a more general statement: the optima of any component-based strictly convex cost function on any subset of L 1 -sphere in N n are precisely the lexicographically minimal elements of this subset. We also present an application in designing optimal CDMA-based wireless sensor networks.

Graph minors and the crossing number of graphs

Abstract There are three general lower bound techniques for the crossing numbers of graphs, all of which can be traced back to Leightons work on applications of crossing number in VLSI: the Crossing Lemma, the Bisection Method, and the Embedding Method. In this contribution, we sketch their adaptations to the minor crossing number.

Long-term survival in glioblastoma: methyl guanine methyl transferase (MGMT) promoter methylation as independent favourable prognostic factor

Abstract Background In spite of significant improvement after multi-modality treatment, prognosis of most patients with glioblastoma remains poor. Standard clinical prognostic factors (age, gender, extent of surgery and performance status) do not clearly predict long-term survival. The aim of this case-control study was to evaluate immuno-histochemical and genetic characteristics of the tumour as additional prognostic factors in glioblastoma. Patients and methods Long-term survivor group were 40 patients with glioblastoma with survival longer than 30 months. Control group were 40 patients with shorter survival and matched to the long-term survivor group according to the clinical prognostic factors. All patients underwent multimodality treatment with surgery, postoperative conformal radiotherapy and temozolomide during and after radiotherapy. Biopsy samples were tested for the methylation of MGMT promoter (with methylation specific polymerase chain reaction), IDH1 (with immunohistochemistry), IDH2, CDKN2A and CDKN2B (with multiplex ligation-dependent probe amplification), and 1p and 19q mutations (with fluorescent in situ hybridization). Results Methylation of MGMT promoter was found in 95% and in 36% in the long-term survivor and control groups, respectively (p < 0.001). IDH1 R132H mutated patients had a non-significant lower risk of dying from glioblastoma (p = 0.437), in comparison to patients without this mutation. Other mutations were rare, with no significant difference between the two groups. Conclusions Molecular and genetic testing offers additional prognostic and predictive information for patients with glioblastoma. The most important finding of our analysis is that in the absence of MGMT promoter methylation, longterm survival is very rare. For patients without this mutation, alternative treatments should be explored.

General Lower Bounds for the Minor Crossing Number of Graphs

There are three general lower bound techniques for the crossing numbers of graphs: the Crossing Lemma, the bisection method and the embedding method. In this contribution, we present their adaptations to the minor crossing number. Using the adapted bounds, we improve on the known bounds on the minor crossing number of hypercubes. We also point out relations of the minor crossing number to string graphs and establish a lower bound for the standard crossing number in terms of Randič index.

On Degree Properties of Crossing-Critical Families of Graphs

Answering an open question from 2007, we construct infinite k-crossing-critical families of graphs which contain vertices of any prescribed odd degree, for sufficiently largei¾?k. From this we derive that, for any set of integers D such that

Nonparametric algorithm for identification of outliers in environmental data: Nonparametric algorithm for identification of outliers

\min D\ge 3

A characterization of plane Gauss paragraphs

and