Abstract
The cell rotation graph D( G) on the strongly connected orientations of a 2-edge-connected plane graph G is defined. It is shown that D( G) is a directed forest and every component is an in-tree with one root; if T is a component of D( G), the reversions of all orientations in T induce a component of D( G), denoted by T −, thus ( T, T −) is called a pair of in-trees of D( G); G is Eulerian if and only if D( G) has an odd number of components (all Eulerian orientations of G induce the same component of D( G)); the width and height of T are equal to that of T −, respectively. Further it is shown that the pair of directed tree structures on the perfect matchings of a plane elementary bipartite graph G coincide with a pair of in-trees of D( G). Accordingly, such a pair of in-trees on the perfect matchings of any plane bipartite graph have the same width and height.
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