Abstract
By the vertex connectivity \(k(G)\) of a graph \(G\) we mean the smallest number of vertices whose removal results in a disconnected or trivial graph, i.e., a graph with one vertex. By the edge connectivity \(\lambda(G)\) of a nontrivial graph \(G\) we mean the smallest number of edges whose removal results in a disconnected graph. The vertex connectivity \(k(G)\), the edge connectivity \(\lambda(G)\), and the minimum vertex degree \(\delta(G)\) of a graph are related by the following inequality found by Whitney in 1932: \(k(G) \leq \lambda(G) \leq \delta(G)\). More recently, Chartrand and Harary proved that for any positive integers \(a\), \(b\), and \(c\) such that \(0 < a \leq b \leq c\) there exists a graph \(G\) such that \(k(G)=a\), \(\lambda(G)=b\), and \(\delta(G)=c\). In the proof, one constructs a family of such graphs with \(2(c+1)\) vertices and \(c(c+1)+b\) edges each. However, it is easily seen that one can sometimes construct a suitable graph with fewer vertices and edges. For example, if \(a=b=c\), then the complete graph with \(c+ 1\) vertices is optimal. In the present paper, we consider the problem of minimizing the Chartrand–Harary construction. We show that for \(b=c\), it is possible to construct a graph with fewer vertices. Four cases are considered separately: \(a=b=c\), \(a<b<c\), \(a=b<c\), and \(a<b=c\). For each case, we present graphs with the prescribed characteristics \(k(G)=a\), \(\lambda(G)=b\), and \(\delta(G)=c\) and with minimum numbers of vertices and edges.
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References
F. Harary, Graph Theory (Addison–Wesley, Reading, MA, 1969).
A. M. Bogomolov and V. N. Salii, Algebraic Foundations of the Theory of Discrete Systems (Nauka, Moscow, 1997), [in Russian].
H. Whitney, “Congruent graphs and the connectivity of graphs,” Amer. J. Math. 54 (1), 150–168 (1932).
G. Chartrand and F. Harary, “Graphs with prescribed connectivities,” in Theory of Graphs, Proc. Colloq., Tihany, 1966 (Academic Press, New York, 1968), pp. 61–63.
B. A. Terebin and M. B. Abrosimov, “On optimal realizations of graphs with prescribed connectivities,” in Applied Discrete Mathematics. Applications (Isd. Dom TGU, Tomsk, 2020), pp. 103–105 [in Russian].
B. A. Terebin and M. B. Abrosimov, “On the minimum number of edges in the realizations of graphs with prescribed connectivities,” in Computer Sciences and Information Technologies (Nauka, Saratov, 2021), pp. 159–161 [in Russian].
Funding
This work was supported by the Ministry of Science and Higher Education of the Russian Federation (grant no. FSRR-2020-0006).
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Translated from Matematicheskie Zametki, 2023, Vol. 113, pp. 323–331 https://doi.org/10.4213/mzm13517.
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Abrosimov, M.B., Terebin, B.A. Optimal Graphs with Prescribed Connectivities. Math Notes 113, 319–326 (2023). https://doi.org/10.1134/S000143462303001X
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DOI: https://doi.org/10.1134/S000143462303001X