Related papers: Vertex links and the Grushko decomposition
We give algorithms with running time $2^{O({\sqrt{k}\log{k}})} \cdot n^{O(1)}$ for the following problems. Given an $n$-vertex unit disk graph $G$ and an integer $k$, decide whether $G$ contains (1) a path on exactly/at least $k$ vertices,…
A previously proposed two-step algorithm for calculating the expectation values of Chern-Simons graphs fails to determine certain crucial signs. The step which involves calculating tetrahedra by solving certain non- linear equations is…
Nagel and R\"omer introduced the class of weakly vertex decomposable simplicial complexes, which include matroid, shifted, and Gorenstein complexes as well as vertex decomposable complexes. They proved that the Stanley-Reisner ideal of…
Graph clustering is an important algorithmic technique for analysing massive graphs, and has been widely applied in many research fields of data science. While the objective of most graph clustering algorithms is to find a vertex set of low…
Given a finite graph G there is a corresponding group given by the presentation with generators the vertices of G and a relation [x,y]=1 for generators x and y precisely when (x,y) is an edge of G. Such groups are known as partially…
Sparse-dense partitions was introduced by Feder, Hell, Klein, and Motwani [STOC 1999, SIDMA 2003] as a tool to solve partitioning problems. In this paper, the following result concerning independent sets in graphs having sparse-dense…
A (unit) disk graph is the intersection graph of closed (unit) disks in the plane. Almost three decades ago, an elegant polynomial-time algorithm was found for \textsc{Maximum Clique} on unit disk graphs [Clark, Colbourn, Johnson; Discrete…
A graph is said to be well-covered if all its maximal independent sets are of the same size. In 1999, Yamashita and Kameda introduced a subclass of well-covered graphs, called localizable graphs and defined as graphs having a partition of…
In this paper, we reveal an intriguing relationship between two seemingly unrelated notions: letter graphs and geometric grid classes of permutations. An important property common for both of them is well-quasi-orderability, implying, in a…
A major open question in the theory of Gorenstein liaison is whether or not every arithmetically Cohen--Macaulay subscheme of $\mathbb{P}^n$ can be G-linked to a complete intersection. Migliore and Nagel showed that, if such a scheme is…
We consider the fundamental problems of determining the rooted and global edge and vertex connectivities (and computing the corresponding cuts) in directed graphs. For rooted (and hence also global) edge connectivity with small integer…
We consider geodesically convex optimization problems involving distances to a finite set of points $A$ in a CAT(0) cubical complex. Examples include the minimum enclosing ball problem, the weighted mean and median problems, and the…
We introduce graphical complexes of groups, which can be thought of as a generalisation of Coxeter systems with 1-dimensional nerves. We show that these complexes are strictly developable, and we equip the resulting Basic Construction with…
Expander decompositions of graphs have significantly advanced the understanding of many classical graph problems and led to numerous fundamental theoretical results. However, their adoption in practice has been hindered due to their…
We give a necessary and sufficient condition for the fundamental group of a finite graph of groups with infinite cyclic edge groups to be acylindrically hyperbolic, from which it follows that a finitely generated group splitting over Z…
Understanding natural phenomenon through the interactions of different complex systems has become an increasing focus in scientific inquiry. Defining complexity and actually measuring it is an ongoing debate and no standard framework has…
This thesis deals with the conjugacy problem in groups and its twisted variants. We analyze recent results by Bogopolski, Martino, Maslakova and Ventura on the twisted conjugacy problem in free groups and its implication for the conjugacy…
Let $G$ be a group of permutations acting on an $n$-vertex set $V$, and $X$ and $Y$ be two simple graphs on $V$. We say that $X$ and $Y$ are $G$-isomorphic if $Y$ belongs to the orbit of $X$ under the action of $G$. One can naturally…
A novel and detailed convergence analysis is presented for a greedy algorithm that was previously introduced for operator reconstruction problems in the field of quantum mechanics. This algorithm is based on an offline/online decomposition…
Dimer coverings (or perfect matchings) of a finite graph are classical objects of graph theory appearing in the study of exactly solvable models of statistical mechanics. We introduce more general dimer labelings which form a topological…