P versus NP is considered as one of the most fundamental open problems in computer science. This consists in knowing the answer of the following question: Is P equal to NP? It was essentially mentioned in 1955 from a letter written by John Nash to the United States National Security Agency. However, a precise statement of the P versus NP problem wa...
The P versus NP problem is a fundamental question in computer science. It asks whether problems whose solutions can be quickly verified can also be quickly solved. Here, "quickly" refers to computational time that grows proportionally to the size of the input (polynomial time). While the problem's roots trace back to a 1955 letter from John Nash, i...
Iskanje najkrajših poti z enim virom v uteženih grafih je klasičen problem na področju algoritmov in podatkovnih struktur. Ta problem lahko rešujemo z znanim Dijkstrovim algoritmom. V diplomski nalogi pa obravnavamo različico problema, pri kateri se v vsakem koraku lahko spremeni teža neke povezave grafa, lahko pa se povezava tudi doda ali odstrani...
This thesis focuses on combinatorial reconfiguration and structural problems with ageometric flavor. Reconfiguration is the study of configuration spaces with respectto an elementary operation transforming a configuration into another. Configurationspaces are formalized through the reconfiguration graph, whose vertices arethe configurations and in ...
One of the biggest unsolved mysteries in computer science is the P versus NP problem. It asks a simple question: can every problem whose solution can be quickly verified be solved just as quickly (Here, "quickly" means in polynomial time)? While the question itself was hinted at in a 1955 letter from John Nash, a formalization of the problem is cre...
We initiate the study of computing shortest non-separating simple closed curves with some given topological properties on non-orientable surfaces. While, for orientable surfaces, any two non-separating simple closed curves are related by a self-homeomorphism of the surface, and computing shortest such curves has been vastly studied, for non-orienta...
In this paper we consider the discrete Laplacian acting on 1-forms and we study its spectrum relative to the spectrum of the 0-form Laplacian. We show that the non zero spectrum can coincide for these Laplacians with the same nature. We examine the characteristics of 0-spectrum of the 1-form Laplacian compared to the cycles of graphs.
Published in Journal of Physics: Conference Series
The problem of signed domination of graphs is a typical optimization problem. It requires that each vertex and edge be assigned a feasible label so that the sum of assigned labels is minimized. First, we propose the concept of a signed total mixed Roman domination number and introduce the related research progress. Finally, we determine several low...
This dissertation contains various combinatorial results about games, graphs and finite Abelian groups. A linear configuration is said to be *common* in an Abelian group $G$ if every 2-colouring of $G$ yields at least as many monochromatic instances of the configuration as a randomly chosen colouring. In Chapter 2, we show that every configuration ...
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