Abstract
AbstractA graph is called ‐connected if is ‐edge‐connected and is ‐edge‐connected for every vertex . The study of ‐connected graphs is motivated by a theorem of Thomassen [J. Combin. Theory Ser. A 110 (2015), pp. 67–78] (that was a conjecture of Frank [SIAM J. Discrete Math. 5 (1992), no. 1, pp. 25–53]), which states that a graph has a ‐vertex‐connected orientation if and only if it is (2,2)‐connected. In this paper, we provide a construction of the family of ‐connected graphs for even, which generalizes the construction given by Jordán [J. Graph Theory 52 (2006), pp. 217–229] for (2,2)‐connected graphs. We also solve the corresponding connectivity augmentation problem: given a graph and an integer , what is the minimum number of edges to be added to make ‐connected. Both these results are based on a new splitting‐off theorem for ‐connected graphs.
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