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<sub>G-F</sub>(p) + deg<sub>G-F</sub>(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<sup>g, d, k</sup>, is the minimum cardinality of a conditional (g, d, k)-cut. Based on these requirements, we obtain K<sup>1,1,k</sup> and K<sup>1,d,1</sup> for hyper cubes.

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