## Graph Theory Lecture Notes 4

### Digraphs (reaching)

**Def**: path
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.

### Graphs (joining)

**Def**: chain
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

**Def**: (connected) component