The Enumeration of Flippable Edges in Maximal Planar Graphs

Much attention has been paid to the diagonal flips in maximal planar graphs. In this paper, we firstly focus on the properties of the unflippable edges in maximal planar graphs, and propose the concept of K-4-embedding. Also, we prove that for a maximal planar graph G with order n(>= 5), an edge e is an element of E(G) is unflippable if and only if e is either incident to a 3-degree vertex or a supporting edge of a K-4-embedding. Secondly, we give the necessary and sufficient condition for a maximal planar graph G has a given number of flippable edges. Finally, we show a general algorithm of the enumeration of flippable edges in maximal planar graphs by the extending-contracting operational system, with the time complexity of O(n(2)).