Conditional Connectivity with Distance Requirement on Hypercube

We introduce a new measure of conditional connectivity. Let G be a connected graph. A set of nodes F is said to be a conditional (g, d, k)-cut of G if (1) G-F is disconnected, (2) every node in G-F has at least g good neighbors, (3) deg_G-F(p) + deg_G-F(q) ≥ 2g + k for every two distinct nodes p and q in G-F with d(p, q) ≤ d. The (g, d, k)-conditional-connectivity, denoted by K^{g, d, k}, is the minimum cardinality of a conditional (g, d, k)-cut. Based on these requirements, we obtain K^1,1,k and K^1,d,1 for hyper cubes.