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A connected graph is one which has all nodes connected by edges, as follows:
The above graph is connected because all nodes are linked to each other by edges, i.e. no islands.
A path traverses from one node to another, without repetition. For example:
In the previous drawing, ABC is a path.
If the starting node and the ending node are the same, the path is also a cycle.
Here, ADCBA is a cycle.
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When a cycle visits ALL the nodes of the graph without repetition, it is a Hamiltonian cycle.
In the above example, ABCDE is a Hamiltonian cycle, and ADCBA is also a Hamiltonian cycle.
On the other hand, if a PATH visits all the nodes on the graph (but does not return to the original node), then it is a hamiltonian path.
Note: some synonyms used (probably in different countries) could be:
cycle = tour
path = trail
For a list of downloadable books on Graph Theory, try: