Disjunkte Fragmente in kritisch n-fach zusammenhängenden Graphen
A graph G is called critically n-connected if G is n-connected and for any vertex x of G the induced subgraph G-\(\{\) \(x\}\) is n-1 connected. Let T be a separating set in G of order n then any union of components of G-T is called a fragment of G. The author proved that every finite, non- complete, critically n-connected graph contains two disjoint fragments F such that \(| F| \leq n/2\). This implies the results of \textit{Y. O. Hamidoune} [Discrete Math. 32, 257-262 (1980; Zbl 0452.05043)] and \textit{H. J. Veldman} [Discrete Math. 44, 105-110 (1983; Zbl 0542.05043)] that each such graph encloses two vertices of degree \(\leq (3/2)n-1\). Secondly the author shows that there are at least four disjoint fragments of any order in such a graph. This improves a result of \textit{L. Nebeský} [J. Graph Theory 1, 69-74 (1977; Zbl 0359.05030)] on the existence of 4 vertices of degree \(\geq 2\) in a critically 2-connected graph.
scientific article; zbMATH DE number 3948317