Verification on a Given Point Set for a Cubic Plane Graph

Abstract

Cubic graph, where all vertices have degree three can be associated as 3-regular graph and trivalent graph. Let a point P be the given point set, with a stipulation of n ≥ 4 points in the regular plane such that n is even in general situation. It has been probed that the problem whether there is a straight-line cubic plane graph on P. Algorithm is not known for polynomialtime of a problem. With research on the reduction to the existence of the convex hull of P along with the certain diagonals of the boundary cycle, the polynomial-time algorithm is proposed that checks for 2-connected cubic plane graphs. The algorithm is constructive and runs in time complexity of O (n3). It is also shown that which graph structure can be expected when there is a cubic plane graph on P; e. g., a cubic plane graph on P implies a connected cubic plane graph on P, and a 2-connected cubic plane graph on P implies a 2-connected cubic plane graph on P that contains the boundary cycle of P.