Abstract
It is proved that every non-complete, finite digraph of connectivity number k has a fragment F containing at most k critical vertices. The following result is a direct consequence: every k-connected, finite digraph D of minimum out- and indegree at least \(2k+ m- 1\) for positive integers k, m has a subdigraph H of minimum outdegree or minimum indegree at least \(m-1\) such that \(D - x\) is k-connected for all \(x \in V(H)\). For \(m = 1\), this implies immediately the existence of a vertex of indegree or outdegree less than 2k in a k-critical, finite digraph, which was proved in Mader (J Comb Theory (B) 53:260–272, 1991).
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References
Halin, R.: A theorem on n-connected graphs. J. Comb. Theory 7, 150–154 (1969)
Halin, R.: Zur Theorie der n-fach zusammenhängenden Graphen. Abh. Math. Sem. Univ. Hambg. 33, 133–164 (1969)
Halin, R.: Untersuchungen über minimale n-fach zusammenhängende Graphen. Math. Ann. 182, 175–188 (1969)
Halin, R.: Kreise beschränkter Länge in gewissen minimalen n-fach zusammenhängenden Graphen. Math. Ann. 183, 323–327 (1969)
Halin, R.: Studies on minimally n-connected graphs. In: Combinatorial Mathematics and its Applications, pp. 129–136. Academic Press, London (1971)
Halin, R.: Unendliche minimale n-fach zusammenhängende Graphen. Abh. Math. Sem. Univ. Hambg. 36, 75–88 (1971)
Lick, D.R.: Minimally n-line connected graphs. J. Reine Angew. Math. 252, 178–182 (1972)
Chartrand, G., Kaugars, A., Lick, D.R.: Critically n-connected graphs. Proc. Am. Math. Soc. 32, 63–68 (1972)
Kameda, T.: Note on Halin’s theorem on minimally connected graphs. J. Comb. Theory (B) 17, 1–4 (1974)
Mader, W.: Ecken vom Innen- und Außengrad n in minimal n-fach kantenzusammenhängenden Digraphen. Arch. Math. 25, 107–112 (1974)
Hamidoune, Y.O.: Sur les atomes d‘un graphe orienté. C. R. Acad. Sci. Paris Sér. A 284, 1253–1256 (1977)
Mader, W.: Connectivity and edge-connectivity in finite graphs. In: Bolllobás, B. (ed.) Surveys in Combinatorics, pp. 66–95. Cambridge University Press, Cambridge (1979)
Mader, W.: Minimal n-fach zusammenhängende Digraphen. J. Comb. Theory (B) 38, 102–117 (1985)
Mader, W.: Kritisch n-fach kantenzusammenhängende Graphen. J. Comb. Theory (B) 40, 152–158 (1986)
Mader, W.: Generalizations of critical connectivity of graphs. Discret. Math. 72, 267–283 (1988)
Mader, W.: On critically connected digraphs. J. Graph Theory 13, 513–522 (1989)
Mader, W.: Ecken von kleinem Grad in kritisch n-fach zusammenhängenden Digraphen. J. Comb. Theory (B) 53, 260–272 (1991)
Mader, W.: Critically n-connected digraphs. In: Alavi, Y., Chartrand, G., Oellermann, O.R., Schwenk, A.J. (eds.) Graph Theory, Combinatorics, and Applications, vol. 2, pp. 811–829. Wiley, New York (1991)
Mader, W.: On vertices of outdegree n in minimally n-connected digraphs. J. Graph Theory 39, 129–144 (2002)
Mader, W.: High connectivity keeping sets in graphs and digraphs. Discret. Math. 302, 173–187 (2005)
Mader, W.: Connectivity keeping trees in k-connected graphs. J. Graph Theory 69, 324–329 (2012)
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Dedicated to the memory of my friend Rudolf Halin.
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Mader, W. Critical vertices in k-connected digraphs. Abh. Math. Semin. Univ. Hambg. 87, 409–419 (2017). https://doi.org/10.1007/s12188-016-0173-y
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DOI: https://doi.org/10.1007/s12188-016-0173-y