Filling of a given boundary by p-gons and related problems Original Research Article

We consider here image-polycycles (image) i.e. plane graphs, such that all interior faces are p-gons, all interior vertices are s-valent and any vertex of the boundary (i.e. the exterior face) has valency within image. The boundary sequence of a image-polycycle P is the sequence image enumerating, up to a cyclic shift or reversal, the consecutive valencies of vertices of the boundary. We show that the values image are the only ones, such that the boundary sequence defines its image-filling (i.e. a image-polycycle with given boundary) uniquely.