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