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Halin graph
In graph theory, a Halin graph is a type of planar graph. It is constructed from a tree that has at least four vertices, none of which have exactly two neighbors. The tree is drawn in the plane so none of its edges cross; then edges are added that connect all its leaves into a cycle.〔''Encyclopaedia of Mathematics'', first Supplementary volume, 1988, ISBN 0-7923-4709-9, p. 281, article ("Halin Graph" ), and references therein.〕 Halin graphs are named after German mathematician Rudolf Halin, who studied them in 1971,〔.〕 but the cubic Halin graphs had already been studied over a century earlier by Kirkman.〔 == Construction ==
A Halin graph is constructed as follows. Let be a tree with more than three vertices, such that no vertex of has degree two (that is, no vertex has exactly two neighbors), embedded in the plane. Then a Halin graph is constructed by adding to a cycle through each of its leaves, such that the augmented graph remains planar.
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