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In this note we gather the theoretical outlines of three basic algorithms for tangles in abstract separation systems: a naive tree search for finding tangles; an algorithm which outputs a certificate for the non-existence of tangles if…

In this note we compute some enumerative invariants of real and complex projective spaces by means of some enriched graphs called floor diagrams.

We present a complete classification of spherical knotoids with up to six crossings and conjecture that our classification up to seven crossings is complete. Our work extends the tradition of knot tabulation to the setting of knotoids…

We specify the computational complexity of crosscap numbers of alternating knots by introducing an automatic computation. For an alternating knot $K$, let $\cal{E}$ be the number of edges of its diagram. Then there exists a code such that…

We show a generalization of the crossing lemma for multi-graphs drawn on orientable surfaces in which pairs of edges are assumed to be drawn by non-homotopic simple arcs which pairwise cross at most $k$ times.

K-fold cross-validation is a widely used tool for assessing classifier performance. The reproducibility crisis faced by artificial intelligence partly results from the irreproducibility of reported k-fold cross-validation-based performance…

I present an algorithm that, given a number $n \geq 1$, computes a compact representation of the set of all noncrossing acyclic digraphs with $n$ nodes. This compact representation can be used as the basis for a wide range of dynamic…

Finding all maximal $k$-plexes on networks is a fundamental research problem in graph analysis due to many important applications, such as community detection, biological graph analysis, and so on. A $k$-plex is a subgraph in which every…

We address the question of counting maps between projective spaces such that images of cycles on the source intersect cycles on the target. In this paper we do it by embedding maps into quasimaps that form a projective space of their own.…

Tangles of graphs have been introduced by Robertson and Seymour in the context of their graph minor theory. Tangles may be viewed as describing "k-connected components" of a graph (though in a twisted way). They play an important role in…

In extension problems of partial graph drawings one is given an incomplete drawing of an input graph $G$ and is asked to complete the drawing while maintaining certain properties. A prominent area where such problems arise is that of…

Computational topology is an area that revisits topological problems from an algorithmic point of view, and develops topological tools for improved algorithms. We survey results in computational topology that are concerned with graphs drawn…

We introduce a theory of multigraded Cayley-Chow forms associated to subvarieties of products of projective spaces. Two new phenomena arise: first, the construction turns out to require certain inequalities on the dimensions of projections;…

Given a subset of active nodes in a network can we re- construct the cascade that has generated these observa- tions? This is a problem that has been studied in the literature, but here we focus in the case that tempo- ral information is…

We introduce the family of $k$-gap-planar graphs for $k \geq 0$, i.e., graphs that have a drawing in which each crossing is assigned to one of the two involved edges and each edge is assigned at most $k$ of its crossings. This definition is…

We study plane algebraic curves defined over a field k of arbitrary characteristic as coverings of the the projective line and the problem of enumerating branched coverings of $\mathbb{P}^{1}$ by using combinatorial methods.

We study the enumeration problem of higher dimensional partitions, a natural generalisation of classical integer partitions. We show that their counting problem is equivalent to the enumeration of simpler classes of higher dimensional…

Topological drawings are natural representations of graphs in the plane, where vertices are represented by points, and edges by curves connecting the points. Topological drawings of complete graphs and of complete bipartite graphs have been…

We introduce and study a new graph representation where vertices are embedded in three or more dimensions, and in which the edges are drawn on the projections onto the axis-parallel planes. We show that the complete graph on $n$ vertices…

The study of nonplanar drawings of graphs with restricted crossing configurations is a well-established topic in graph drawing, often referred to as beyond-planar graph drawing. One of the most studied types of drawings in this area are the…