Polycycles with given boundary
Michel Marie Deza, Mathieu Dutour Sikirić · Cambridge University Press eBooks · 2008
The ( r, q ) -boundary sequence of a finite ( r, q )-polycycle P is the sequence b ( P ) of numbers enumerating, up to a cyclic shift or reversal, the consecutive degrees of vertices incident to the exterior face. For earlier applications of this (and other) codes, see. Given an ( r, q )-boundary sequence b , a plane graph P is called a ( r, q ) -filling of b if P is an ( r, q )-polycycle such that b = b ( P ). In this chapter we consider the unicity of those ( r, q )-fillings and algorithms used for their computations. The problem of uniqueness of ( r, q )-fillings By inspecting the list of ( r, q )-polycycles for ( r, q ) = (3, 3), (3, 4), or (4, 3) in Section 4.2, we find that the ( r, q )-boundary sequence of an ( r, q )-polycycle determines it uniquely. We expect that for any other pair ( r, q ) this is not so We show that the value r = 3, 4 are the only ones, such that the ( r , 3)-boundary sequence always defines its ( r , 3)-filling uniquely. Note that an ( r, q )-polycycle, which is not an unique filling of its boundary, is, necessarily, a helicene. Some examples of non-uniqueness of ( r , 3)-fillings are cases of boundaries b , admitting an ( r , 3)-filling P with the symmetry group of b being larger than the symmetry group of P , implying the existence of several different ( r , 3)-fillings.