Two points are said to be connected (form a connection) when there are no other points along the line between them. Thus a single point is trivially connected.
A path is a set of connected points. There are two kinds of paths. A simple path is a set of connected points which form a straight line. A compound path is a set of simple paths, called sub-paths, such that each sub-path is connected to at least one other sub-path within the compound path.
A stone is said to be adjacent to another stone when it occupies a point that is connected to the point occupied by the first stone. When stones of the same color are adjacent to one another they are said to be connected.
Just as it is with points, a single stone is trivally connected. A group of stones can be connected along a simple or compound path. A simple connected group of stones is a set of connected stones lying on a simple path. The simple connected group ends at the edge of the board or when there is a connected point that is not occupied by a stone of the same color (it is an unoccupied point or is occupied by a stone of the other color). A compound connected group of stones is a set of stones lying on a compound path where all of the stones are connected.