An Exact Enumeration of Distance-Hereditary Graphs

Distance-hereditary graphs form an important class of graphs, from the theoretical point of view, due to the fact that they are the totally decomposable graphs for the split-decomposition. The previous best enumerative result for these graphs is from Nakano et al. (J. Comp. Sci. Tech., 2007), who have proven that the number of distancehereditary graphs on n vertices is bounded by 2d3.59ne. In this paper, using classical tools of enumerative combinatorics, we improve on this result by providing an exact enumeration and full asymptotic of distance-hereditary graphs, which allows to show that the number of distancehereditary graphs on n vertices is tightly bounded by (7.24975 . . .)n—opening the perspective such graphs could be encoded on 3n bits. We also provide the exact enumeration and full asymptoticss of an important subclass, the 3-leaf power graphs. Our work illustrates the power of revisiting graph decomposition results through the framework of analytic combinatorics. Introduction The decomposition of graphs into tree-structures is a fundamental paradigm in graph theory, with algorithmic and theoretical applications [5]. In the present work, we are interested in the split-decomposition, introduced by Cunningham and Edmonds [9, 10] and recently revisited by Gioan et al. [20, 21, 7]. For the classical modular and split-decomposition, the decomposition tree of a graph G is a tree (rooted for the modular decomposition and unrooted for the split decomposition) of which the leaves are in bijection with the vertices of G and whose internal nodes are labeled by indecomposable (for the chosen decomposition) graphs; such trees are called graph-labeled trees by Gioan and Paul [20]. Moreover, there is a one-to-one correspondence between such trees and graphs. The notion of a graph being totally decomposable for a decomposition scheme translates into restrictions on the labels that can ap∗Dept. of Mathematics, Simon Fraser University, 8888 University Drive, V5A 1S6, Burnaby (BC), Canada, [email protected] †CNRS & LIX, École Polytechnique, 91120 Palaiseau, France, [email protected] ‡Dept. of Computer Science, Princeton University, 35 Olden Street, Princeton, NJ 08540, USA, [email protected] pear on the internal nodes of its decomposition tree. For example, for the split-decomposition, totally decomposable graphs are the graphs whose decomposition tree’s internal nodes are labeled only by cliques and stars; such graphs are called distance-hereditary graphs. They generalize the wellknown cographs, the graphs that are totally decomposable for the modular decomposition, and whose enumeration has been well studied, in particular by Ravelomanana and Thimonier [26], also using techniques from analytic combinatorics Efficiently encoding graph classes1 is naturally linked to the enumeration of such graph classes. Indeed the number of graphs of a given class on n vertices implies a lower bound on the best possible encoding one can hope for. Until recently, few enumerative properties were known for distance-hereditary graphs, unlike their counterpart for the modular decomposition, the cographs. The best result so far, by Nakano et al. [24], relies on a relatively complex encoding on 4n bits, whose detailed analysis shows that there are at most 2b3.59nc unlabeled distance-hereditary graphs on n vertices. However, using the same techniques, their result also implies an upper-bound of 2 for the number of unlabeled cographs on n vertices, which is far from being optimal for these graphs, as it is known that, asymptotically, there are Cdn/n3/2 such graphs where C = 0.4126 . . . and d = 3.5608 . . . [26]. This suggests there is room for improving the best upper bound on the number of distancehereditary graphs provided by Nakano et al. [24], which was the main purpose of our present work. This paper. Following a now well established approach, which enumerates graph classes through a tree representation, when available (see for example the survey by Giménez and Noy [19] on tree-decompositions to count families of planar graphs), we provide combinatorial specifications, in the sense of Flajolet and Sedgewick [17], of the split-decomposition trees of distance-hereditary graphs and 3-leaf power graphs, both in the labeled and unlabeled cases. From these specifications, we can provide exact enumerations, asymptotics, and leave open the possibility of uniform random samplers allowing for further empirical studies of 1By which we mean, describing any graph from a class with as few bits as possible, as described for instance by Spinrad [28]. statistics on these graphs (see Iriza [23]). In particular, we show that the number of distancehereditary graphs on n vertices is bounded from above by 23n, which naturally opens the question of encoding such graphs on 3n bits, instead of 4n bits as done by Nakano et al. [24]. We also provide similar results for 3leaf power graphs, an interesting class of distance hereditary graphs, showing that the number of 3-leaf power graphs on n vertices is bounded from above by 22n. Main results. Our main contribution is to introduce the idea of symbolically specifying the trees arising from the splitdecomposition, so as to provide the (previously unknown) exact enumeration of certain important classes of graphs. Our grammars for distance-hereditary graphs are in Subsection 3, and our grammars for 3-leaf power graphs are in Subsection 2. We provide here the corollary that gives the beginning of the exact enumerations for the unlabeled and unrooted versions of both classes2. Corollary 1 (Enumeration of connected, unlabeled, unrooted distance-hereditary graphs). The first few terms of the enumeration, EIS A277862, are 1, 1, 2, 6, 18, 73, 308, 1484, 7492, 40010, 220676, 1253940, 7282316, 43096792, 259019070, 1577653196, 9720170360, 60492629435 . . . and the asymptotics is c · 7.249751250 . . .n · n−5/2 with c ≈ 0.02337516194 . . .. Corollary 2 (Enumeration of connected, unlabeled, unrooted 3-leaf power graphs). The first few terms of the enumeration, EIS A277863, are 1, 1, 2, 5, 12, 32, 82, 227, 629, 1840, 5456, 16701, 51939, 164688, 529070, 1722271, 5664786, 18813360, 62996841, 212533216 . . . and the asymptotics is c · 3.848442876 . . .n · n−5/2 with c ≈ 0.70955825396 . . .. 1 Definitions and Preliminaries For a graph G, we denote by V (G) its vertex set and E(G) its edge set. Moreover, for a vertex x of a graphG, we denote by N(x) the neighbourhood of x, that is the set of vertices y ∈ V (G) such that {x, y} ∈ E(G); this notion extends naturally to vertex sets: if V1 ⊆ V (G), then N(V1) is the set of vertices in V (G)\V1 that is adjacent to at least one vertex 2With the symbolic grammars, it is then easy to establish recurrences [18, 29] to efficiently compute the enumeration–to the extent that we were trivially able to obtain the first 10 000 terms of the enumerations. See a survey by Flajolet and Salvy [16, §1.3] for more detail. in V1. Finally, the subgraph of G induced by a subset V1 of vertices is denoted by G[V1]. A graph on n vertices is labeled if its vertices are identified with the set {1, . . . , n}, with no two vertices having the same label. A graph is unlabeled if its vertices are indistinguishable. A clique on k vertices, denoted Kk is the complete graph on k vertices (i.e., there exists an edge between every pair of vertices). A star on k vertices, denoted Sk, is the graph with one vertex of degree k− 1 (the center of the star) and k − 1 vertices of degree 1 (the extremities of the star). 1.1 Split-decomposition trees. We first introduce the notion of graph-labeled tree, due to Gioan and Paul [20], then define the split-decomposition and the corresponding tree, described as a graph-labeled tree. Definition 1. A graph-labeled tree (T,F) is a tree3 T in which every internal node v of degree k is labeled by a graph Gv ∈ F on k vertices, such that there is a bijection ρv from the edges of T incident to v to the vertices of Gv . Definition 2. A split [9] of a graph G with vertex set V is a bipartition (V1, V2) of V (i.e., V = V1 ∪ V2, V1 ∩ V2 = ∅) such that (a) |V1| > 1 and |V2| > 1; (b) every vertex of N(V1) is adjacent to every of N(V2). A graph without any split is called a prime graph. A graph is degenerate if any partition of its vertices without a singleton part is a split: cliques and stars are the only such graphs. Informally, the split-decomposition of a graph G consists in finding a split (V1, V2) in G, followed by decomposing G into two graphs G1 = G[V1 ∪ {x1}] where x1 ∈ N(V1) andG2 = G[V2∪{x2}] where x2 ∈ N(V2) and then recursively decomposing G1 and G2. This decomposition naturally defines an unrooted tree structure of which the internal vertices are labeled by degenerate or prime graphs and whose leaves are in bijection with the vertices of G, called a split-decomposition tree. A split-decomposition tree (T,F) with F containing only cliques with at least three vertices and stars with at least three vertices is called a cliquestar tree. It can be shown that the split-decomposition tree of a graph might not be unique (i.e., that several decompositions sequences of a given graph can lead to different splitdecomposition trees), but following Cunningham [9], we obtain the following uniqueness result, reformulated in terms of graph-labeled trees by Gioan and Paul [20]. Theorem (Cunningham [9]). For every connected graph G, there exists a unique split-decomposition tree such that: 3This is a non-plane tree: the ordering of the children of an internal node does not matter—this is why in most of our grammars we describe the children as a SET instead of a SEQ, a sequence.