A connected component of a node is the biggest subgraph of a graph that consists of all the nodes that serve as endpoints of walks starting at the given node. A connected component of one node is a connected component of all its nodes. The graph of the Konigsberg Bridges is not simple. Obviously, a graph is connected if and only if it consists of a single connected component. This is more or less obvious that the board of the 15 puzzle might be abstracted to a 4x4 graph every counter position corresponding to a node.
Edges indicate possible puzzle moves, i.e. moves of the empty square.
Less obvious is that the graph is bipartite. There is nothing to prepare you for this fact. Even as I write this I feel a tinge of surprise. Power may be a wrong word or be wasted fruitlessly unless a benefit is gained by looking at the puzzle as a game on a graph. And a benefit there is as I plan to demonstrate shortly.