Counting Outerplanar Maps

A map is outerplanar if all its vertices lie in the outer face. We enumerate various classes of rooted outerplanar maps with respect to the number of edges and vertices. The proofs involve several bijections with lattice paths. As a consequence of our results, we obtain an efficient scheme for encoding simple outerplanar maps. 1 Statement of results A map is a connected planar multigraph with a specific embedding in the 2-sphere, up to oriented homeomorphisms [5]. All maps we consider are rooted: a root edge is selected and oriented in one of the two possible directions. By convention, when represented in the plane the drawing is such that the face to the right of the oriented root edge is the outer face. The root vertex is the tail of the root edge. A rooted map is outerplanar if all its vertices are in the outer face. Our main goal is to count outerplanar maps according to the number of edges or vertices, subject to various conditions. The main tool is to set up bijections with several kinds of lattice paths. A Dyck path is a lattice path from (0, 0) to (2n, 0) using steps U = (1, 1) and D = (1,−1) that never goes below the x axis. A peak in a Dyck path is a sequence UD, and a valley is a sequence DU . Considering a Dyck path as a well-formed system of parentheses, every U step has its matching D step. It is well known that the number of Dyck paths of length 2n is the Catalan number Cn = 1 n+1 ( 2n n ) . We also deal with Dyck paths in which every down step can either be marked or not (equivalently, can be coloured in one of two colours). Clearly the number of such paths is 2Cn. A Schröder path is a lattice path from (0, 0) to (2n, 0) using steps U , D and H = (2, 0), that never goes below the x axis. A Schröder path is small if it has no H steps at level zero. ∗Partly supported by grant MTM2014-54745-P. the electronic journal of combinatorics 24(2) (2017), #P2.3 1 The number of Schröder paths of length 2n is the Schröder number rn. It is well-known that the number sn of small Schröder paths satisfies 2sn = rn, for n > 1 (see [4]). We also need the following result: it seems to be folklore, but not being able to find a suitable reference we provide a short proof of it. Lemma 1. The number of small Schröder paths of length 2n with m horizontal steps is equal to 1 n ( n m )( 2n−m n+ 1 ) , 0 6 m 6 n− 1. (1) In this paper we prove the following results, which to the best of our knowledge are new. A map is loopless if it has no loops, and is simple if it is loopless and has no multiple edges. Theorem 2. The number of outerplanar maps with n edges is equal to 2Cn, for n > 1. The number of outerplanar maps with n edges and k vertices is equal to ( n k − 1 ) Cn = 1 n+ 1 ( n k − 1 )( 2n n ) , 1 6 k 6 n+ 1. Theorem 3. The number of loopless outerplanar maps with n edges is equal to the small Schröder number sn. The number of loopless outerplanar maps with n edges and k vertices is equal to 1 k − 1 ( n− 1 k − 2 )( n+ k − 1 k − 2 ) , 2 6 k 6 n+ 1. Theorem 4. The number of simple outerplanar maps with n edges and k vertices is equal to 1 n ( n k − 1 )( 2k − 2 n+ 1 ) , k − 1 6 n 6 2k − 3. The extreme values for n are Catalan numbers and correspond to trees (n = k − 1) and to polygon triangulations (n = 2k − 3). Finally we show a bijection between simple outerplanar maps and certain Dyck paths. Theorem 5. There is a bijection between simple outerplanar maps and the set of Dyck paths in which every U step at non-zero level can be marked or not. The bijection takes simple outerplanar maps with n edges and k vertices to Dyck paths of length 2(k− 1) with n− k + 1 marked steps. As a consequence of the previous result we obtain an optimal encoding for simple outerplanar maps having k vertices with 3k bits, 2k bits for encoding the Dyck path and k additional bits for marking the U steps. Such an encoding was obtained earlier in [2] Sequence A033282 in OEIS Sequence A033282 in OEIS, called Borel’s triangle. the electronic journal of combinatorics 24(2) (2017), #P2.3 2 using a rather involved argument, based on selecting a special spanning tree of the map and encoding the additional edges. Our encoding is much simpler and easier to compute. It is known [1] that the number of (not embedded) connected unlabelled outerplanar graphs with k vertices grows (up to a polynomial term) like γ, where γ ≈ 7.5. It follows that (log2 γ)k ≈ 2.91 · k bits are needed in any case to encode such a graph. Hence our scheme provides an encoding with 3k bits of unlabelled connected outerplanar graphs by fixing a particular embedding, which is close from being optimal. We notice that outerplanar graphs have been studied too from a metric point point of view: it is shown in [3] that, scaled by a factor 1/ √ n, a random planar map converges to Aldous’ Continuum Random Tree.