A classification of outerplanar K-gonal 2-trees

Essentially, a 2-tree is a simple connected graph composed by triangles glued along their edges in a tree-like fashion, that is, without cycles (of triangles). The enumeration of 2-trees has been extensively studied in the literature. See, for instance, Harary and Palmer [8] and Fowler et al [6]. Here we consider more general 2-trees, where the triangles are replaced by quadrilaterals, pentagons, or polygons with K sides (K-gons), K ≥ 3. The term K-gonal 2-trees is used when K is fixed, triangular, quadrangular, pentagonal, . . . , for K = 3, 4, 5, . . . , respectively, and polygonal 2-trees when the polygon size K is allowed to vary. The labelled, unlabelled and asymptotic enumeration of K-gonal 2-trees for any fixed K is considered in [12] and in [13], where the perimeter is taken into account. The K-gonal 2-trees are not to be confused with K-dimensional trees, or K-trees, which are built with K-simplices glued together along (K − 1)-faces in a tree-like fashion. Formulas have been given in [2] and [5] for the labelled enumeration of K-trees but their unlabelled enumeration is still an open problem. A graph is called outerplanar if it can be embedded in the plane in such a way that every vertex lies on the outer face. Their unlabelled and asymptotic enumeration has been recently carried out by Bodirsky, Fusy, Kang and Vigerske [3]. It is easily seen that 2-connected outerplanar graphs can be identified with polygonal 2-trees. Figure 1 shows an example of two pentagonal 2-trees, the first one being outerplanar but not the second one. Notice that the degree of any edge (that is, the number of K-gons incident to it) of an outerplanar K-gonal 2-tree cannot exceed 2. The edge is called internal if it is of degree 2 and external otherwise. The enumeration of outerplanar K-gonal 2-trees has been studied by Harary, Palmer and Read [15, 9] in connection with the cell growth problem and dissections of a polygon. These structures are also of interest in combinatorial chemistry since for K = 6, for example, they correspond to special classes of catacondensed benzenoid hydrocarbons (see Gutman and Cyvin [7]). Other values of K, for example, 3, 4, 5, 7 and 8 have also been considered in the chemical literature. The goal of the present work is to give a more refined classification of unlabelled outerplanar K-gonal 2-trees, according to their symmetries, extending previous work by Labelle et al. [11] for ordinary (that is, where K = 3) outerplanar 2-trees, in the framework of the combinatorial theory of species of Joyal [10, 1]. Our classification is closely related to the symmetry nomenclature used by chemists in the case of plane molecules and is expressed as the molecular expansion of the corresponding species.