Conditional graph connectivity relative to hereditary properties

A graphical property P is said to be hereditary (strongly hereditary) if every induced subgraph (subgraph) of a graph with property P also has property P . If P is a graphical property, then the P ‐connectivity of a graph is the minimum number of vertices whose removal from G produces the trivial graph or a disconnected graph each of whose components has property P . Several analogs and generalizations of results concerning the ordinary connectivity of a graph are established for the P ‐connectivity of a graph, with respect to hereditary properties P . If P is a graphical property, then the P ‐edge‐connectivity of a graph is defined similarly to the P ‐connectivity. Several results concerning the P ‐edge‐connectivity of a graph with respect to strongly hereditary properties P are established. Moreover, a generalization of Whitney's inequalities is given.