# Кафедра математики

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### Browsing Кафедра математики by Author "Kozerenko, Sergiy"

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Item All-path convexity: two characterizations, general position number, and one algorithm(2024) Haponenko, Vladyslav; Kozerenko, SergiyWe present two characterizations for the all-path convex sets in graphs. Using the first criterion, we obtain a new characterization of connected block graphs and compute the general position number in a graph with respect to the all-path convexity. The second criterion allows us to provide a new algorithm for testing a set on all-path convexity.Item An optimal lower bound for the size of periodic digraphs(2023) Kozerenko, SergiyA periodic digraph is the digraph associated with a periodic point of a continuous map from the unit interval to itself. This digraph encodes "covering" relation between minimal intervals in the corresponding orbit, which allows the application of purely combinatorial arguments in establishing results on the existence and co-existence of periods of periodic points (for example, in proving the famous Sharkovsky’s theorem). In this article, an optimal lower bound for the size of periodic digraphs is provided and thus some previous results of Pavlenko are tightened.Item Unique eccentric point graphs and their eccentric digraphs(2023) Hak, Artem; Haponenko, Vladyslav; Kozerenko, Sergiy; Serdiuk, AndriiWe study graph-theoretic properties of eccentric digraphs of unique eccentric point graphs (shortly, uep-graphs). The latter are the connected graphs in which every vertex has a unique eccentric vertex. In particular, we characterize uep-graphs and the corresponding eccentric digraphs in the following classes: self-centered graphs having the number of vertices twice as diameter, block graphs, and graphs with diameter three. Also, we obtain non-trivial properties of weak components in eccentric digraphs of uep-graphs with diameter four and pose several open questions in this direction.