András Pluhár

Covering planar graphs with forests

We study the problem of covering graphs with trees and a graph of bounded maximum degree. By a classical theorem of Nash-Williams, every planar graph can be covered by three trees. We show that every planar graph can be covered by two trees and a forest, and the maximum degree of the forest is at most 8. Stronger results are obtained for some special classes of planar graphs.

The accelerated k -in-a-row game

We investigate two versions of the well-known k-in-a-row game. While in the most intriguing k=5 case the outcome of the game has been recently settled, very little is known about what happens when the rules are changed. A natural modification is that the players take more than one square of the board per move in order to speed up the game. Our main goal is to improve the quadratic bound on the error term, given by Csirmaz in Csirmaz (Discrete Math. 29 (1980) 19-23), to a logarithmic one for the accelerated k-in-a-row. The other issue is the extreme sensitivity of k-in-a-row under biased rules. Beck proposed in Beck (unpublished lecture notes) that a player may trade some of his freedom of choice for the right of taking more squares than his opponent. We prove logarithmic bounds on the error term in that case, too.

On Chooser-Picker positional games

Two new versions of the so-called Maker-Breaker Positional Games are defined by Jozsef Beck. In these variants Picker takes unselected pair of elements and Chooser keeps one of these elements and gives back the other to Picker. In the Picker-Chooser version Picker is Maker and Chooser is Breaker, while the roles are swapped in the Chooser-Picker version. It seems that both the Picker-Chooser and Chooser-Picker versions are not worse for Picker than the original Maker-Breaker versions. Here we give winning conditions for Picker in some Chooser-Picker games that extend the results of Beck.

Community detection by using the extended modularity

This article is about community detection algorithms in graphs. First a new method will be introduced, which is based on an extension [16] of the commonly used modularity [17, 18, 19, 20] and gives overlapping communities. We list and compare the results given by our new method and some other algorithms yileding either overlapping or non-overlapping communities. While the main use of the proposed algorithm is benchmarking, we also consider the possibility of hot starts, and some further extensions that considers the degree distribution of the graphs.

Approximations of the Generalized Cascade Model

The study of infection processes is an important field of science both from the theoretical and the practical point of view, and has many applications. In this paper we focus on the popular Independent Cascade model and its generalization. Unfortunately the exact computation of infection probabilities is a #P-complete problem [8], so one cannot expect fast exact algorithms. We propose several methods to efficiently compute infection patterns with acceptable accuracy. We will also examine the possibility of substituting the Independent Cascade model with a computationally more tractable model.

A sharp edge bound on the interval number of a graph

A main result proved in this paper is the following. Theorem. Let G be a noncomplete graph on n vertices with degree sequence d1 ≥ d2 ≥ · · · ≥ dn and t ≥ 2 be a prime. Let m = gcd{t, di - dj: 1 ≤ i < j ≤ n} and set

Applications of the inverse infection problem on bank transaction networks

d =\cases{1\ \ \ if\ m = t\ and\ \ m \not\mid\ d_{i}\ for\ 1 \leq i \leq n \cr 0\ \ \ otherwise.}

Dynamic communities and their detection

Then R(tG, ℤt) = t(n + d) - d, where R is the zero-sum Ramsey number. This settles, almost completely, problems raised in [Bialostocki & Dierker, J Graph Theory, 1994; Y. Caro, J Graph Theory, 1991].