arXiv subjects

Aneta Pokorná

Publications and source records attributed to Aneta Pokorná.

6 recordsLinked to original sources

arXiv2025

Reconstructing graphs and their connectivity using graphlets

Graphlets are subgraphs rooted at a fixed vertex. The number of occurrences of graphlets aligned to a particular vertex, called graphlet degree sequence (gds), gives a topological description of the surrounding of the analyzed vertex. Graphlet degree distribution (gdd) of a graph is a matrix containing graphlet degree sequence for all vertices in the given graph. A long standing open problem called reconstruction conjecture (RC) asks whether the structure of a graph is uniquely determined by the multiset of its vertex-deleted subgraphs. Graphlet degree distribution up to size (n - 1), (<= n - 1)-gdd, gives more information to reconstruct the graph and we use it to reconstruct any graph having a unique almost-asymmetric vertex-deleted subgraph, where almost-asymmetric means that at most one automorphism orbit has size larger than one. Moreover, we prove that any graph containing a vertex-cut of size 1 or any graph of order n having a vertex with degree at most 2 or at least n-2 is reconstructible from its (<= n - 1)-gdd, which expands results shown in the standard RC. We also discuss the relation between gdd and graph connectivity and the conditions on (<= 3)-gdd, whose breaking means that no graph with such gdd exists.

math.CO↗

arXiv2023

Structure of betweenness uniform graphs with low values of betweenness centrality

This work deals with undirected graphs that have the same betweenness centrality for each vertex, so-called betweenness uniform graphs (or BUGs). The class of these graphs is not trivial and its classification is still an open problem. Recently, Gago, Coroničová-Hurajová and Madaras conjectured that for every rational $α\ge 3/4$ there exists a BUG having betweenness centrality~$α$. We disprove this conjecture, and provide an alternative view of the structure of betweenness-uniform graphs from the point of view of their complement. This allows us to characterise all the BUGs with betweennes centrality at most 9/10, and show that their betweenness centrality is equal to $\frac{\ell}{\ell+1}$ for some integer $\ell\le 9$. We conjecture that this characterization extends to all the BUGs with betweenness centrality smaller than~1.

math.CO↗

arXiv2023

Computing approximate symmetries of complex networks

The symmetry of complex networks is a global property that has recently gained attention since MacArthur et al. 2008 showed that many real-world networks contain a considerable number of symmetries. These authors work with a very strict symmetry definition based on the network's automorphism. The potential problem with this approach is that even a slight change in the graph's structure can remove or create some symmetry. Recently, Liu 2020 proposed to use an approximate automorphism instead of strict automorphism. This method can discover symmetries in the network while accepting some minor imperfections in their structure. The proposed numerical method, however, exhibits some performance problems and has some limitations while it assumes the absence of fixed points. In this work, we exploit alternative approaches recently developed for treating the Graph Matching Problem and propose a method, which we will refer to as Quadratic Symmetry Approximator (QSA), to address the aforementioned shortcomings. To test our method, we propose a set of random graph models suitable for assessing a wide family of approximate symmetry algorithms. The performance of our method is also demonstrated on brain networks.

cs.SI↗

arXiv2023

On the Connectivity and the Diameter of Betweenness-Uniform Graphs

Betweenness centrality is a centrality measure based on the overall amount of shortest paths passing through a given vertex. A graph is betweenness-uniform if all its vertices have the same betweenness centrality. We study the properties of betweenness-uniform graphs. In particular, we show that every connected betweenness-uniform graph is either a cycle or a $3$-connected graph. Also, we show that betweenness uniform graphs of high maximal degree have small diameter.

math.CO↗

arXiv2021

Constructions of betweenness-uniform graphs from trees

Betweenness centrality is a measure of the importance of a vertex x inside a network based on the fraction of shortest paths passing through x. We study a blow-up construction that has been shown to produce graphs with uniform distribution of betweenness. We disprove the conjecture about this procedure's universality by showing that trees with a diameter at least three cannot be transformed into betweenness-uniform by the blow-up construction. It remains open to characterize graphs for which the blow-up construction can produce betweenness-uniform graphs.

math.CO↗

arXiv2021

On 3-Coloring of $(2P_4,C_5)$-Free Graphs

The 3-coloring of hereditary graph classes has been a deeply-researched problem in the last decade. A hereditary graph class is characterized by a (possibly infinite) list of minimal forbidden induced subgraphs $H_1,H_2,\ldots$; the graphs in the class are called $(H_1,H_2,\ldots)$-free. The complexity of 3-coloring is far from being understood, even for classes defined by a few small forbidden induced subgraphs. For $H$-free graphs, the complexity is settled for any $H$ on up to seven vertices. There are only two unsolved cases on eight vertices, namely $2P_4$ and $P_8$. For $P_8$-free graphs, some partial results are known, but to the best of our knowledge, $2P_4$-free graphs have not been explored yet. In this paper, we show that the 3-coloring problem is polynomial-time solvable on $(2P_4,C_5)$-free graphs.

cs.DS↗