Final answer:
A path that begins and ends at the same vertex and passes through all other vertices exactly once is called a Hamiltonian circuit, which is the correct option (b) for the given question.
Step-by-step explanation:
A path in a graph that starts and ends at the same vertex and passes through all other vertices exactly once is called a Hamiltonian circuit.
The term Hamiltonian circuit is named after the Irish mathematician William Rowan Hamilton who studied such paths in the context of the Icosian Game or Hamilton's puzzle. Unlike Euler circuits which require that all edges be visited once, Hamiltonian circuits focus on visiting each vertex once and returning to the starting vertex.
The correct option for the given question is (b) Hamiltonian. It's important to differentiate between various types of paths or circuits in graph theory, as they each have unique properties and applications. A closed path refers to any path that starts and ends at the same vertex, but does not have the requirement of visiting every other vertex exactly once.
Euler paths or circuits involve covering every edge of the graph once, while a bipartite graph is a type of graph where its vertices can be divided into two disjoint sets such that every edge connects a vertex in one set to a vertex in the other set.