Stability results for uniquely determined sets from two directions in discrete tomography

On uniquely k-determined permutations

There are several approaches to study occurrences of consecutive patterns in permutations such as the inclusion-exclusion method, the tree representations of permutations, the spectral approach and others. We propose yet another approach to study occurrences of consecutive patterns in permutations. The approach is based on considering the graph of patterns overlaps, which is a certain subgraph of the de Bruijn graph. While applying our approach, the notion of a uniquely $k$-determined permutation appears. We give two criteria for a permutation to be uniquely $k$-determined: one in terms of the…

A simplicial complex is uniquely determined by its set of discrete Morse functions

We prove that a connected simplicial complex is uniquely determined by its complex of discrete Morse functions. This settles a question raised by Chari and Joswig. In the 1-dimensional case, this implies that the complex of rooted forests of a connected graph G completely determines G.

On the difference between solutions of discrete tomography problems II

We consider the problem of reconstructing binary images from their horizontal and vertical projections. It is known that the projections do not necessarily determine the image uniquely. In a previous paper it was shown that the symmetric difference between two solutions (binary images that satisfy the projections) is at most 4A times the square root of 2N. Here N is the sum of the projections in one direction (i.e. the size of the image) and A is a parameter depending on the projections. In this paper we give a lower bound: for each set of projections that has at least two solutions, we constr…

The Farey graph is uniquely determined by its connectivity

We show that, up to minor-equivalence, the Farey graph is the unique minor-minimal graph that is infinitely edge-connected but such that every two vertices can be finitely separated.

离散断层成像中的误差校正

Error Correction for Discrete Tomography

本文研究了在离散断层成像中线和存在少量错误时的误差校正问题，证明了可以纠正少于d/2个错误，并证明该界是最优的。

Canonical Horizontal Visibility Graphs are uniquely determined by their degree sequence

Horizontal visibility graphs (HVGs) are graphs constructed in correspondence with number sequences that have been introduced and explored recently in the context of graph-theoretical time series analysis. In most of the cases simple measures based on the degree sequence (or functionals of these such as entropies over degree and joint degree distributions) appear to be highly informative features for automatic classification and provide nontrivial information on the associated dynam- ical process, working even better than more sophisticated topological metrics. It is thus an open question why t…