A path is simple if all of its vertices are distinct.
A path is closed if the first vertex is the same as the last vertex (i.e., it starts and ends at the same vertex.)
A cycle is a simple closed path.
Note: a cycle is not a simple path. Also, all the arcs are distinct.
In the above digraph,
2 - 9 - 8 - 10 - 11 - 9 - 8 - 7 is a path (neither simple nor closed)
1 - 4 - 6 - 5 - 7 is a path which is simple.
9 - 8 - 10 - 11 - 9 - 10 - 11 - 9 is a path which is closed.
1 - 2 - 3 - 4 - 1 is a cycle.
We say that vertex u can be reached from vertex v if there is a path from v to u. Note that this is not a symmetric relationship.
A chain is simple if all of its vertices are distinct.
A chain is closed if the first vertex is the same as the last vertex.
A circuit is a simple closed chain with all of its edges distinct.
In the above graph,
1 - 2 - 3 - 4 - 2 is a chain (neither simple nor closed).
6 - 7 - 8 - 9 is a simple chain.
1 - 2 - 3 - 4 - 2 - 1 is a closed chain.
6 - 7 - 8 - 5 - 6 is a circuit.
We say that a vertex u is joined to a vertex v if there is a chain from v to u. This is a symmetric relationship.
The length of a path (or chain) is the number of arcs (resp. edges) in the path (resp. chain). This number is one less than the number of vertices.
Def: complete graph, complete symmetric digraph
Def: strongly connected (digraph), connected (graph)
Def: Subgraph, induced (generated) subgraph