Sparse certificates and removable cycles in l-mixed p-connected graphs

A graph G = (V,E) is called l-mixed p-connected if G−S−L is connected for all pairs S, L with S ⊆ V , L ⊆ E, and l|S|+ |L| < p. This notion is a common generalisation of m-vertex-connectivity (l = 1, p = m) and m-edge-connectivity (l ≥ m, p = m). If p = kl then we obtain (k, l)-connectivity, introduced earlier by Kaneko and Ota, as a special case. We show that by using maximum adjacency orderings one can find sparse local certificates for l-mixed p-connectivity in linear time, provided the maximum edge multiplicity is at most l. A by-product of this result is a short proof for the existence of (and a linear time algorithm to find) a cycle C in an l-mixed p-connected graph with minimum degree at least p + 2, for which G − E(C) is l-mixed p-connected. This extends a result of Mader on removable cycles in k-vertex-connected graphs.