Found the answers. Sorry for not researching properly.

The definition of simple graph is given by the text book Discrete Mathematics with Applications second edition by Susanna S. Epp from page 609:

"A simple graph is a graph that does not have any loops or parallel edges."

And the definition of -connected graph is given by k-Connected Graph -- from Wolfram MathWorld

"A graph is said to be -connected (or -vertex connected, or -point connected) if there does not exist a set of vertices whose removal disconnects the graph, i.e., the vertex connectivity of is (Skiena 1990, p. 177). Therefore, a connected graph is -connected, and a biconnected graph is -connected."