Abstract
With graphs considered as natural models for many network design problems, edge connectivity κ′(G) and maximum number of edge-disjoint spanning trees τ(G) of a graph G have been used as measures for reliability and strength in communication networks modeled as graph G (see Cunningham, in J ACM 32:549–561, 1985; Matula, in Proceedings of 28th Symposium Foundations of Computer Science, pp 249–251, 1987, among others). Mader (Math Ann 191:21–28, 1971) and Matula (J Appl Math 22:459–480, 1972) introduced the maximum subgraph edge connectivity \({\overline{\kappa'}(G) = {\rm max} \{\kappa'(H) : H {\rm is} \, {\rm a} \, {\rm subgraph} \, {\rm of} G \}}\) . Motivated by their applications in network design and by the established inequalities
we present the following in this paper:
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1.
For each integer k > 0, a characterization for graphs G with the property that \({\overline{\kappa'}(G) \le k}\) but for any edge e not in G, \({\overline{\kappa'}(G + e) \ge k+1}\).
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2.
For any integer n > 0, a characterization for graphs G with |V(G)| = n such that κ′(G) = τ(G) with |E(G)| minimized.
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Gu, X., Lai, HJ., Li, P. et al. Characterizations of Strength Extremal Graphs. Graphs and Combinatorics 30, 1453–1461 (2014). https://doi.org/10.1007/s00373-013-1359-z
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DOI: https://doi.org/10.1007/s00373-013-1359-z