Abstract
Let κ(G) denote the (vertex) connectivity of a graph G. For ℓ≥0, a noncomplete graph of finite connectivity is called ℓ-critical if κ(G−X)=κ(G)−|X| for every X⊆V(G) with |X|≤ℓ.
Mader proved that every 3-critical graph has diameter at most 4 and asked for 3-critical graphs having diameter exceeding 2. Here we give an affirmative answer by constructing an ℓ-critical graph of diameter 3 for every ℓ≥3.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Diestel, R.: Graph Theory. Graduate Texts in Mathematics 173, Springer, New York (2000)
Mader, W.: Endlichkeitssätze für k-kritische Graphen. Math. Ann. 229, 143–153 (1977)
Mader, W.: On k-critically n-connected graphs. Progress in Graph Theory, Academic Press, pp. 389–398 (1984)
Mader, W.: On k-con-Critically n-connected Graphs. J. Combin. Theory (B) 86, 296–314 (2002)
Maurer, St. B., Slater, P.J.: On k-critical n-connected graphs. Discrete Math. 20, 255–262 (1977)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kriesell, M. There Exist Highly Critically Connected Graphs of Diameter Three. Graphs and Combinatorics 22, 481–485 (2006). https://doi.org/10.1007/s00373-006-0672-1
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s00373-006-0672-1