A path is a sequence of vertices in which consecutive vertices are joined by edges.

A path provides a route through the graph. Its length is the number of edges it uses.

A cycle is a closed path that returns to its starting vertex without repeating intermediate vertices.

Connectedness

An undirected graph is connected when every pair of vertices is joined by some path.

If a graph is not connected, its vertices split into connected components: maximal groups whose vertices are mutually reachable.

Paths, cycles and components describe global structure that cannot be read from one edge at a time. They are the basic language for reachability, networks and later graph algorithms.