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Vestnik Yuzhno-Ural'skogo Gosudarstvennogo Universiteta. Seriya "Matematika. Mekhanika. Fizika", 2021, Volume 13, Issue 4, Pages 77–89
DOI: https://doi.org/10.14529/mmph210409
(Mi vyurm503)
 

Short communications

The minimal vertex extensions for colored complete graphs

P. V. Razumovsky, M. B. Abrosimov

Saratov State University, Saratov, Russian Federation
References:
Abstract: The article proposes the results of the search for minimal vertex extensions of undirected colored complete graphs. The research topic is related to the modelling of full fault tolerant technical systems with a different type of their objects in the terminology of graph theory. Let a technical system be Σ, then there is a graph G(Σ), which vertices reflects system's objects and edges reflects connections between these objects. Type of each object reflected in a mapping of some color from F={1,2,i} to the corresponding vertex. System's Σ vertex extension is a graph G(Σ) which contains additional vertices. System reflected by graph G(Σ) can work even if there are k faults of its objects. Complete graph is a graph where each two vertices have an edge between them. Complete graphs have no edge extensions because there is no way to add additional edge to the graph with a maximum number of edges. In other words, the system reflected by some complete graph cannot be able to resist connection faults. Therefore the article research is focused on vertex extensions only. There is a description of vertex extensions existence condition for those colored complete graphs. This paper considers generating schemes for such minimal vertex extensions along with formulas, which allows to calculate number of additional edges to have an ability to construct minimal vertex extension.
Keywords: graph vertex extensions, complete graphs, graph minimal extensions, colored graph extensions, colored graphs.
Funding agency Grant number
Ministry of Science and Higher Education of the Russian Federation FSRR-2020-0006
Received: 28.10.2021
Document Type: Article
UDC: 519.17
Language: Russian
Citation: P. V. Razumovsky, M. B. Abrosimov, “The minimal vertex extensions for colored complete graphs”, Vestn. Yuzhno-Ural. Gos. Un-ta. Ser. Matem. Mekh. Fiz., 13:4 (2021), 77–89
Citation in format AMSBIB
\Bibitem{RazAbr21}
\by P.~V.~Razumovsky, M.~B.~Abrosimov
\paper The minimal vertex extensions for colored complete graphs
\jour Vestn. Yuzhno-Ural. Gos. Un-ta. Ser. Matem. Mekh. Fiz.
\yr 2021
\vol 13
\issue 4
\pages 77--89
\mathnet{http://mi.mathnet.ru/vyurm503}
\crossref{https://doi.org/10.14529/mmph210409}
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