Hypohamiltonian graph

In the mathematical field of graph theory, a graph G is said to be hypohamiltonian if G does not itself have a Hamiltonian cycle but every graph formed by removing a single vertex from G is Hamiltonian.

Hypohamiltonian graphs were first studied by Sousselier (1963). Lindgren (1967) cites Gaudin, Herz & Rossi (1964) and Busacker & Saaty (1965) as additional early papers on the subject; another early work is by Herz, Duby & Vigué (1967).

Grötschel (1980) sums up much of the research in this area with the following sentence: “The articles dealing with those graphs ... usually exhibit new classes of hypohamiltonian or hypotraceable graphs showing that for certain orders n such graphs indeed exist or that they possess strange and unexpected properties.”

Hypohamiltonian graphs arise in integer programming solutions to the traveling salesman problem: certain kinds of hypohamiltonian graphs define facets of the traveling salesman polytope, a shape defined as the convex hull of the set of possible solutions to the traveling salesman problem, and these facets may be used in cutting-plane methods for solving the problem.Grötschel (1980) observes that the computational complexity of determining whether a graph is hypohamiltonian, although unknown, is likely to be high, making it difficult to find facets of these types except for those defined by small hypohamiltonian graphs; fortunately, the smallest graphs lead to the strongest inequalities for this application.

Concepts closely related to hypohamiltonicity have also been used by Park, Lim & Kim (2007) to measure the fault tolerance of network topologies for parallel computing.

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