Кафедра математики

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All-path convexity: two characterizations, general position number, and one algorithm
(2024) Haponenko, Vladyslav; Kozerenko, Sergiy
We 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.
Unique eccentric point graphs and their eccentric digraphs
(2023) Hak, Artem; Haponenko, Vladyslav; Kozerenko, Sergiy; Serdiuk, Andrii
We 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.