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We resolve in the affirmative conjectures of Repovs and A. Skopenkov (1998), and M. Skopenkov (2003) generalizing the classical Hanani-Tutte theorem to the setting of approximating maps of graphs on 2-dimensional surfaces by embeddings. Our…

The cyclability of a graph is the maximum integer $k$ for which every $k$ vertices lie on a cycle. The algorithmic version of the problem, given a graph $G$ and a non-negative integer $k,$ decide whether the cyclability of $G$ is at least…

Large datasets with interactions between objects are common to numerous scientific fields (i.e. social science, internet, biology...). The interactions naturally define a graph and a common way to explore or summarize such dataset is graph…

Consider a graph drawn on a surface (for example, the plane minus a finite set of obstacle points), possibly with crossings. We provide an algorithm to decide whether such a drawing can be untangled, namely, if one can slide the vertices…

We discuss the problem of embedding graphs in the plane with restrictions on the vertex mapping. In particular, we introduce a technique for drawing planar graphs with a fixed vertex mapping that bounds the number of times edges bend. An…

Let $\mathbb{S}_g$ be the orientable surface of genus $g$. We show that the number of vertex-labelled cubic multigraphs embeddable on $\mathbb{S}_g$ with $2n$ vertices is asymptotically $c_g n^{5(g-1)/2-1}\gamma^{2n}(2n)!$, where $\gamma$…

We study clustering on graphs with multiple edge types. Our main motivation is that similarities between objects can be measured in many different metrics. For instance similarity between two papers can be based on common authors, where…

Graph clustering is the problem of identifying sparsely connected dense subgraphs (clusters) in a given graph. Proposed clustering algorithms usually optimize various fitness functions that measure the quality of a cluster within the graph.…

We study a family of graph clustering problems where each cluster has to satisfy a certain local requirement. Formally, let $\mu$ be a function on the subsets of vertices of a graph $G$. In the $(\mu,p,q)$-PARTITION problem, the task is to…

Spectral clustering is popular among practitioners and theoreticians alike. While performance guarantees for spectral clustering are well understood, recent studies have focused on enforcing ``fairness'' in clusters, requiring them to be…

Face recognition sees remarkable progress in recent years, and its performance has reached a very high level. Taking it to a next level requires substantially larger data, which would involve prohibitive annotation cost. Hence, exploiting…

Using Je\v{r}\'abek 's framework for probabilistic reasoning, we formalize the correctness of two fundamental RNC^2 algorithms for bipartite perfect matching within the theory VPV for polytime reasoning. The first algorithm is for testing…

Time series clustering poses a significant challenge with diverse applications across domains. A prominent drawback of existing solutions lies in their limited interpretability, often confined to presenting users with centroids. In…

In the framework of graph property testing, we study the problem of determining if a graph admits a cluster structure. We say that a graph is $(k, \phi)$-clusterable if it can be partitioned into at most $k$ parts such that each part has…

A graph is NIC-planar if it admits a drawing in the plane with at most one crossing per edge and such that two pairs of crossing edges share at most one common end vertex. NIC-planarity generalizes IC-planarity, which allows a vertex to be…

Determining whether two graphs are structurally identical is a fundamental problem with applications spanning mathematics, computer science, chemistry, and network science. Despite decades of study, graph isomorphism remains a challenging…

Given a family $\mathcal{F}$ of graphs, a graph is \emph{$\mathcal{F}$-subgraph-free} if it has no subgraph isomorphic to a member of $\mathcal{F}$. We present a fixed-parameter linear-time algorithm that decides whether a planar graph can…

In this thesis, we study two problems based on clustering algorithms. In the first problem, we study the role of visual attributes using an agglomerative clustering algorithm to whittle down the search area where the number of classes is…

We consider the problem of spectral clustering under group fairness constraints, where samples from each sensitive group are approximately proportionally represented in each cluster. Traditional fair spectral clustering (FSC) methods…

In this paper, several two-dimensional clustering scenarios are given. In those scenarios, soft partitioning clustering algorithms (Fuzzy C-means (FCM) and Possibilistic c-means (PCM)) are applied. Afterward, VAT is used to investigate the…