Recognizing trees from incomplete decks

For a given graph, the unlabeled subgraphs $G-v$ are called the cards of $G$ and the deck of $G$ is the multiset $\{G-v: v \in V(G)\}$. Wendy Myrvold [Ars Combinatoria, 1989] showed that a non-connected graph and a connected graph both on $n$ vertices have at most $\lfloor \frac{n}{2} \rfloor +1$ cards in common and she found (infinite) families of trees and non-connected forests for which this upper bound is tight. Bowler, Brown, and Fenner [Journal of Graph Theory, 2010] conjectured that this bound is tight for $n \geq 44$. In this article, we prove this conjecture for sufficiently large $n$. The main result is that a tree $T$ and a unicyclic graph $G$ on $n$ vertices have at most $\lfloor \frac{n}{2} \rfloor+1$ common cards. Combined with Myrvold's work this shows that it can be determined whether a graph on $n$ vertices is a tree from any $\lfloor \frac{n}{2}\rfloor+2$ of its cards. Based on this theorem, it follows that any forest and non-forest also have at most $\lfloor \frac{n}{2} \rfloor +1$ common cards. Moreover, we have classified all except finitely many pairs for which this bound is strict. Furthermore, the main ideas of the proof for trees are used to show that the girth of a graph on $n$ vertices can be determined based on any $\frac{2n}{3} +1$ of its cards. Lastly, we show that any $\frac{5n}{6} +2$ cards determine whether a graph is bipartite.Comment: 22 page

The Many Faces of the Mathematical Modeling Cycle

In literature about mathematical modeling a diversity can be seen in ways of presenting the modeling cycle. Every year, students in the Bachelor’s program Applied Mathematics of the Eindhoven University of Technology, after having completed a series of mathematical modeling projects, have been prompted with a simple three-step representation of the modeling cycle. This representation consisted out of 1) problem translation into a mathematical model, 2) the solution to mathematical problem, and 3) interpretation of the solution in the context of the original problem. The students’ task was to detail and complete this representation. Their representations also showed a great diversity. This diversity is investigated and compared with the representations of the students’ teachers. The representations with written explanations of 77 students and 20 teachers are analyzed with respect to the presence of content aspects such as problem analysis, worlds/models/knowledge other than mathematical, verification, validation, communication and reflection at the end of the modeling process. Also form aspects such as iteration and complexity are analyzed. The results show much diversity within both groups concerning the presence or absence of aspects. Validation is present most, reflection least. Only iteration (one is passing the modeling cycle) more than once is significantly more present in the teachers’ group than in the students’ group. While accepting diversity as a natural phenomenon, the authors plea for incorporating all aspects mentioned into mathematical modeling education

Lucas Bunt and the rise of statistics education in the Netherlands

International audienceWe describe the role of Lucas Bunt at the start of the teaching of probability and statistics in the last two years of Dutch secondary schools in the early 1950s. Together with his co-authors, Bunt developed an experimental text which, from the mid-1950s on, became a regular textbook. We further sketch Bunt’s other – mostly international – activities with respect to the curriculum reform movement initiated at the Royaumont Seminar in 1959. Bunt’s experiment can be seen as one of the initiatives related to this reform. Finally, we present what happened with statistics teaching in the Netherlands “after Bunt”

On non-permutation solutions to some two machine flow shop scheduling problems

In this paper, we study two versions of the two machine flow shop scheduling problem, where schedule length is to be minimized. First, we consider the two machine flow shop with setup, processing, and removal times separated. It is shown that an optimal solution need not be a permutation schedule, and that the problem is NP-hard in the strong sense, which contradicts some known results. The tight worst-case bound for an optimal permutation solution in proportion to a global optimal solution is shown to be 3/2. An O(n) approximation algorithm with this bound is presented. Secondly, we consider the two machine flow shop with finite storage capacity. Again, it is shown that there may not exist an optimal solution that is a permutation schedule, and that the problem is NP-hard in the strong sense