Properties of minimally tt-tough graphs

A graph $G$ is minimally $t$-tough if the toughness of $G$ is $t$ and the deletion of any edge from $G$ decreases the toughness. Kriesell conjectured that for every minimally $1$-tough graph the minimum degree $\delta(G)=2$. We show that in every minimally $1$-tough graph $\delta(G)\le\frac{n+2}{3}$. We also prove that every minimally $1$-tough claw-free graph is a cycle. On the other hand, we show that for every $t \in \mathbb{Q}$ any graph can be embedded as an induced subgraph into a minimally $t$-tough graph

The complexity of recognizing minimally tough graphs

A graph is called $t$-tough if the removal of any vertex set $S$ that disconnects the graph leaves at most $|S|/t$ components. The toughness of a graph is the largest $t$ for which the graph is $t$-tough. A graph is minimally $t$-tough if the toughness of the graph is $t$ and the deletion of any edge from the graph decreases the toughness. The complexity class DP is the set of all languages that can be expressed as the intersection of a language in NP and a language in coNP. In this paper, we prove that recognizing minimally $t$-tough graphs is DP-complete for any positive rational number $t$. We introduce a new notion called weighted toughness, which has a key role in our proof

Minimally toughness in special graph classes

Let $t$ be a positive real number. A graph is called $t$-tough, if the removal of any cutset $S$ leaves at most $|S|/t$ components. The toughness of a graph is the largest $t$ for which the graph is $t$-tough. A graph is minimally $t$-tough, if the toughness of the graph is $t$ and the deletion of any edge from the graph decreases the toughness. In this paper we investigate the minimum degree and the recognizability of minimally $t$-tough graphs in the class of chordal graphs, split graphs, claw-free graphs and $2K_2$-free graphs

Color-avoiding percolation in edge-colored Erd\H{o}s-R\'enyi graphs

We study a variant of the color-avoiding percolation model introduced by Krause et al., namely we investigate the color-avoiding bond percolation setup on (not necessarily properly) edge-colored Erd\H{o}s-R\'{e}nyi random graphs. We say that two vertices are color-avoiding connected in an edge-colored graph if after the removal of the edges of any color, they are in the same component in the remaining graph. The color-avoiding connected components of an edge-colored graph are maximal sets of vertices such that any two of them are color-avoiding connected. We consider the fraction of vertices contained in color-avoiding connected components of a given size as well as the fraction of vertices contained in the giant color-avoiding connected component. Under some mild assumptions on the color-densities, we prove that these quantities converge and the limits can be expressed in terms of probabilities associated to edge-colored branching process trees. We provide explicit formulas for the limit of the normalized size of the giant color-avoiding component, and in the two-colored case we also provide explicit formulas for the limit of the fraction of vertices contained in color-avoiding connected components of a given size.Comment: 59 pages + Appendix + List of notation. Added reference to the recent arXiv preprint arXiv:2211.1608

Canteen reform from the perspective of the parents of nursery children in Békéscsaba

It is important to provide sufficient quantity of good quality food for young children. The bases of healthy diet should be learned as a child and nurseries have an important role in shaping the children’s eating habits and forming their taste. This paper summarizes research results about the canteen reform introduced in nurseries on 1st January, 2015. A total of 140 questionnaires were collected from parents or grandparents of the young children attending nurseries in Békéscsaba. We asked them about their pre-reform and post-reform opinions on the children’s food and what they think of the implemented reform. The answers indicated that the parents were satisfied with the nursery caretakers who always gave detailed information about the child’s meal and provided the possibility to discuss problems, most of the parents were well informed about the the nursery canteen reform and the majority agreed with it. Our proposals based on the research are: The parents should have more influence and their opinion should be asked about the catering of their children. Opinions of the nursery caretakers and the catering managers should also be considered, since they spend a lot of time with the young children at the nursery. The changes in the nursery catering should have been implemented more gradually

Diverzitás és inklúzió jellemzői a Pécsi Tudományegyetemen: - egy helyzetfelmérő kutatás tükrében

The study relies on Hungarian and international research that introduce diversification as a tendency in higher education institutions, and in relation to that detail the practise of increasing inclusivity. The situation-revealing research applies the process-oriented model of inclusion that helps the analysis of different student groups, based on the admittance (possibilities to enter), the process (successful advancement) and the outcome (indi-cators of success) at the University of Pécs. The analysis includes students (N: 68,602) and training programs (N: 83,067) between the years 2010-2019, based on data collected from the Neptun database. The background factors of the analysis were the diverse student groups (prioritized groups: students with learning-mental disor-ders, with disadvantaged background, handicapped and international students), the given benefits (scholarship, dormitory) and the status of the training. The results are visualized on a timeline and on faculty level to high-light the decrease of certain student groups (i.e.: those with disadvantaged backgrounds) and the increase of others (international students). The supporting effect of the given benefits became apparent in the advancement in studies. Based on the data of diploma acquisition, it is transparent that prioritized students are less successful if the training is longer – which supports the necessity for inclusive development in the institution. The present study focuses on the most salient tendencies, however the database offers further possibilities for tracking the learning-process of the analysed groups, which could also explain some tendencies introduced above.Tanulmányunk támaszkodik azokra a hazai és nemzetközi vizsgálatokra, melyek bemutatják, miként vált a felsőoktatás jellemzőjévé a diverzitás, és ezzel összefüggésben tárgyalja az inkluzivitás kiépülésének gyakorlatát. Helyzetfeltáró kutatásunk során az inklúzió folyamatelvű modelljét alkalmazva arra voltunk kiváncsiak, hogy mi jellemzi a különböző hallgatói csoportokat a bemenet (bejutási lehetőségek), a folyamat (sikeres előrehaladás) és a kimenet (eredményességi mutatók) szempontjából a Pécsi Tudományegyetemen. Elemzésünk a 2010 és 2019 között képzéssel rendelkező hallgatókra (N = 68 602) és képzésekre (N = 83 067) terjedt ki, és a Neptunban elérhető adatokon alapult. Az elemzés háttérváltozója a diverz tanulói csoportok (előnyben részesítettek: tanulási-pszichés zavarosok, fogyatékosok, hátrányos helyzetűek, továbbá külföldiek), a kapott juttatások (ösztöndíj, kollégium) és az adott képzés státusza. Eredményeink idősorosan és karokra lebontva mutatnak rá a bemenet oldaláról bizonyos hallgatói csoportok visszaszorulására (pl. hátrányos helyzetűek), illetve létszámnövekedésére (külföldiek). Láthatóvá vált a különböző juttatások támogató ereje az előrehaladás során. A diplomaszerzési adatok alapján megállapítható, hogy minél magasabb fokú a képzés, annál hosszabb ideig és annál kevésbé sikeresen teljesítenek a bemenet során előnyben részesített hallgatói csoportok – rámutatva az intézményi inkluzivitás fejlesztésének szükségességére. Jelen kutatás az elemzett adatbázis legszembetűnőbb tendenciáira fókuszált, azonban az adatbázis további lehetőségeket rejt a vizsgált csoportok tanulási életútjának fókuszáltabb nyomon követésére, a fentiekben leírtak mögött meghúzódó okok egy részének feltárására is

Conditions for minimally tough graphs

Katona, Solt\'esz, and Varga showed that no induced subgraph can be excluded from the class of minimally tough graphs. In this paper, we consider the opposite question, namely which induced subgraphs, if any, must necessarily be present in each minimally $t$-tough graph. Katona and Varga showed that for any rational number $t \in (1/2,1]$, every minimally $t$-tough graph contains a hole. We complement this result by showing that for any rational number $t>1$, every minimally $t$-tough graph must contain either a hole or an induced subgraph isomorphic to the $k$-sun for some integer $k \ge 3$. We also show that for any rational number $t > 1/2$, every minimally $t$-tough graph must contain either an induced $4$-cycle, an induced $5$-cycle, or two independent edges as an induced subgraph