A proper orientation of a simple undirected graph $G=(V,E)$ is an orientation $D$ of $E$ such that $d^-_D(u) \neq d^-_D(v)$ for every two adjacent vertices $u$ and $v$, where $d^-_D(u)$ is the number of incoming arcs to $u$ on $D$. The proper orientation number $pon(G)$ of $G$ is the minimum of the maximum indegrees over all its proper orientations.
It is known that a graph always has a proper orientation and that the proper orientation number is at most the maximum degree. The proper orientation number of a tree is known to be at most 4, and for a cactus, at most 7. On the other hand, it is known that computing the proper orientation number is NP-hard even for 4-regular graphs and planar graphs. This paper focuses on tree-depth $td(G)$ as a graph structural parameter and shows that the proper orientation number is above bounded by $2(td-1)$. We then give a $2^{O(td(G)^2)}n$-time algorithm, where $n$ is the number of vertices in $G$.

In many cutting and packing problems the sole objective is to maximise the utilisation of the cutting board, while satisfying the demand of pieces to be cut. When one focuses on optimising the use of raw material, it is very likely that for each board we will have to enter different cutting coordinates. In operations that involve a large number of pieces many boards will be required, so the cutting process can be highly affected when changing boards. When dealing with a heterogeneous mix of pieces, this problem cannot be avoided. However, when the set of pieces to be cut present large repetitions of the same pieces, it would seem reasonable to pay attention to the number of changes in the machine settings, reducing the cutting process time.

In this problem we introduce a new objective in the classic two-dimensional bin packing problem, where on top of maximising the total utilisation, we are also interested in minimising the number of different patterns needed to complete the demand. For this work, a pattern is defined as the coordinates of a set of pieces that will be cut from a bin. Minimising the number of patterns, will result in less changes on the settings of the cutting machine among bins, with a reduction on total cutting times. This variant of the problem has not been widely studied and so far we have only found the work by \cite{Song2014} which uses a column generation model to solve this problem.

To start tackling this problem, we expand the initial work in \cite{Cabo2018} and work with a similar setting based on the glass industry, with irregular pieces that can be freely rotated and reflected. We also start by considering the restriction of separating each piece by means of a guillotine cut, which reduces the solution space and somehow limit the placement of each piece. \cite{Cabo2018} present a beam search strategy where each node represents a complete bin, thus the branch with minimum depth is the one with less bins, or maximum utilisation. The same idea applies in this problem, interpreting the branch depth as the number of changes in machine settings.

We introduce some novelty approaches in terms of the sorting of pieces, considering not just the largest as the first ones to place, but also taking in consideration the number of repetitions of each piece. The fact that pieces are separated by guillotine cuts, translates into irregular shapes where to place pieces after a guillotine cut is performed, thus we also look at the similarity between the angles of the pieces and the angles of each section of the bin to decide which piece to place next. Preliminary test will be shown to demonstrate the efficiency of these new criteria for sorting pieces.

We also provide initial results that confirms that the strategy of dealing with both objectives simultaneously is more effective than using existing heuristics that are mainly design for heterogenous mix of pieces.

3D Phase Unwrapping is a pivotal step in analyzing MRI imagery, where traditional methods are often of a heuristic nature and suboptimal outcomes. Central to this challenge is the phenomenon of phase values wrapped between 0 and 2pi, which can veil true phase differences and introduce inaccuracies, particularly in critical applications like medical imaging and materials science.

Our study introduces a novel approach that constructs a graph linking neighboring residues, enabling the rapid detection of "Singular Loops" by the use of the "Graph Untangling" method — surpassing the efficiency of existing state-of-the-art methods. We address these loops by identifying their minimal surfaces, and that by solving a discrete version of the "Plateau Problem". This
involves the use of Integer Programming with constraint separation to compute
these minimal surfaces. We here use cycle basis theories to optimize our constraints separation methods, aiming to optimize the unwrapping process.

Subsequently, we reconstruct the original phase while strategically avoiding these Singular Loops, thereby bypassing the identified optimal minimal surfaces. This methodology not only enhances the accuracy of phase reconstruction but also contributes to a deeper understanding of the complexities involved in 3D Phase Unwrapping. Our findings offer a significant advancement in the field, hopefully transforming practices in medical imaging analysis and beyond.

We investigate uncrossed multiflows, multiflows on planar graphs with the property that the curves identified by their flow paths do not cross in the plane. We demonstrate that given a fractional such multiflow, it can be rounded to an integral multiflow of congestion 2, motivating them as an interesting combinatorial object. We explore the complexity of computing uncrossed multiflows. On the one hand we can prove that determining whether a fractional uncrossed multiflow exists is NP-hard. In contrast, an integral (uncrossed) multiflow can be computed efficiently when the number of faces with demands is bounded.