Types and boundary uniqueness of polypentagons

A connected plane graph taken with its interior faces forms an (r, q)-polycycle if the interior faces are all r-gons (where r = const > 3), while all interior vertices are of the maximal possible degree, denoted by q (where q = const > 3); see [1]. Any finite (r, q)-polycycle is homeomorphic to a disc and its boundary is homeomorphic to a circle. The cyclic sequence q1q2 . . . qm formed by the degrees qi, i = 1, . . . , m, of all m boundary vertices of an (r, q)-polycycle is called the boundary code. We call an r-gon near-boundary if at least one of its edges belongs to the boundary. If the intersection of a near-boundary r-gon with the boundary is connected, then the r-gon is removable, that is, the graph without this r-gon remains an (r, q)-polycycle; see [2]. Two (r, q)-polycycles with the same boundary are called equiboundary. Equiboundary (r, q)-polycycles consist of the same number pr of r-gons; see [3]. A pair of equiboundary (r, q)-polycycles is said to be irreducible if every boundary edge belonging to a removable r-gon of one of the (r, q)-polycycles belongs to an unremovable r-gon of the other (r, q)-polycycle; see [2]. An example of an equiboundary pair of (r, 3)-polycycles with the number pr = 4r is given in [4] for any r > 5; it is conjectured there that 4r is the minimal value for pr. The conjecture is proved in [2] for r = 6. Below we prove the conjecture for r = 5, that is, for (5, 3)-polycycles; we call such polycycles polypentagons.