Connected graph A graph G is connected if given any vertices u and v in G, there is a path from u to v (or v to u). That is, in a connected graph, we can get from any vertex to any other vertex on a path. If there’s no path between some pair of vertices then the graph is called disconnected. The following shows a connected graph and a disconnected graph. Example: a b c b a c
e d connected graph
e d disconnected graph
Cycle A cycle (or a circuit) is a path of non-zero length from u to v with no repeated edges. A simple cycle is cycle from v to v, in which there are no repeated vertices (except for the initial and final vertices which are both equal to v) Example: a b d e c (a, b, d, c, b, e, c, a) is a cycle (but not simple) (a, d, c, b, e, a) is a simple cycle of length 5.
Notice that a simple path of length n contains (n+1) distinct vertices, which a simple cycle of length n contains n distinct vertices.
Eulerian graphs A cycle in a connected graph G that includes all the edges and all the vertices of G is called an Euler cycle (“Euler” is pronounced as “oiler”).
A connected graph containing an Euler cycle is called an Eulerian graph.
Finding an Euler cycle in a graph is the same as playing the following “game”. Draw the entire graph without lifting your pen which drawing all the edges exactly once, and visiting all the vertices, starting from and ending at the same vertex. Example: Try playing the “game” described above to show that graphs (1) – (5)is the following graphs are Eulerian. Graph (6) is not Eulerian because you can’t draw the entire graph starting and ending with the same vertex.
Special Types of Graphs There are certain types of graphs to which we give special names. 1. 2. 3. 4. 5. Trivial graph – a graph of order 1 and size 0. Null graph – a graph of order n and size 0; denoted by Nn. Cycle graph – a graph of...