The structure and unlabelled enumeration of toroidal graphs with no K 3,3 s

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1 The structure and unlabelled enumeration of toroidal graphs with no K 3,3 s Andrei Gagarin, Gilbert Labelle, Pierre Leroux 3 Laboratoire de Combinatoire et d Informatique Mathématique (LaCIM) Université du Québec à Montréal (UQAM) Montréal, Québec, Canada Abstract We characterize the toroidal graphs with no K 3,3 -subdivisions as canonical compositions in which -pole planar networs are substituted for the edges of non-planar cores. This structure enables us to enumerate these graphs. We describe an explicit enumerative approach that requires unlabelled enumeration of -connected planar graphs. Keywords: toroidal graph, embedding in a surface, unlabelled enumeration. Introduction We are interested in simple non-planar graphs that can be embedded on the torus or the projective plane. By Kuratowsi s theorem, a graph G is nonplanar if and only if it contains a subdivision of K 5 or K 3,3, and, by Wagner s theorem, a graph G is non-planar if and only if it has a minor isomorphic to K 5 or K 3,3. The graphs with no K 3,3 -subdivisions coincide with the graphs with no K 3,3 -minors. Therefore they are referred to as graphs with no K 3,3 s. gagarin@lacim.uqam.ca labelle.gilbert@uqam.ca 3 leroux.pierre@uqam.ca

2 We describe the structure of -connected non-planar projective-planar and toroidal graphs with no K 3,3 s (denoted respectively by P and T ) in terms of a special substitutional operation G N, where -pole networs of a class N are substituted for the edges of core graphs from a class G. This structure enables us to do the labelled enumeration of graphs in P and T (see [3] and [4]). Here we present the unlabelled enumeration of the same graphs, which is described in detail in [5]. Our enumerative approach is based on Walsh s method [7] rigorized and extended by developing and using general techniques of the theory of species []. Decomposition and structural results A -pole networ (or simply a networ) is a connected graph N with two distinguished vertices 0 and, such that the graph N 0 is -connected (N ab means the graph obtained from N by adding the edge ab if it is absent). The vertices 0 and are called the poles of N, and all the other vertices of N are internal. We define an operator τ acting on -pole networs, N τ N, which interchanges poles 0 and. A class N of networs is called symmetric if N N implies τ N N. A networ N is planar if the graph N 0 is planar. Denote by N P the class of planar networs. The substitution of a -pole networ N for an edge e = uv of a graph G is the set of one or two graphs obtained by replacing the edge e of G with the networ N by identifying the poles of N with the extremities of e (the set of internal vertices of N is disjoint from G). Given a graph G 0 with edges, E = {e,e,...,e }, and a sequence (N,N,...,N ) of networs with disjoint underlying sets, the composition G 0 (N,N,...,N ) is the set of graphs that can be obtained by substituting the networ N j N for the edge e j of G 0, j =,,...,. The graph G 0 is called the core, and the N i s are called components of the resulting graphs. For a class of graphs G and a class of networs N, we denote by G N the class of graphs obtained as compositions G 0 (N,N,...,N ) with G 0 G and N i N, i =,,...,. The composition G N is canonical if for any graph G G N there is a unique core G 0 G and unique (up to pole interchanges) components N,N,...,N N that yield G. Graphs M and M are depicted in Figure. A networ obtained from K 5 by selecting two poles 0 and among the vertices and by removing the edge 0 is called a K 5 \e-networ (see Figure (i)). Denote by C i, i 3, a simple cycle on i vertices. A crown H of K 5 \e-networs is a graph obtained from a cycle C i, i 3, by substituting K 5 \e-networs for some edges of C i in

3 such a way that no two unsubstituted edges of C i are adjacent in H (e.g., see Figure (ii)). Denote by H the class of crowns of K 5 \e-networs. A toroidal core is either K 5, an M-graph, an M -graph, or a crown of K 5 \e-networs. Denote by T C the class of toroidal cores, i.e. T C = K 5 + M + M + H. The following structure theorem is obtained by refining structural and algorithmic results of []. (i) Fig.. (i) The graph M, (ii) The graph M. (ii) 0 (i) Fig.. (i) K 5 \e-networ, (ii) Crown of K 5 \e-networs. Theorem. ([3], [4]) A -connected non-planar graph G with no K 3,3 s is (i) projective-planar if and only if G K 5 N P, and the composition P = K 5 N P is canonical; (ii) toroidal if and only if G T C N P, and the composition T = T C N P is canonical. This characterization enables us to detect toroidal or projective-planar graphs with no K 3,3 s in linear time by analogy with algorithms of [], i.e. by using a breadth-first or depth-first search for the decomposition and by doing a linear-time planarity testing. The linear-time complexity of such an algorithm follows from the linear-time complexity of the decomposition and from the fact that each vertex of the initial graph can appear in at most seven different components. 3 Walsh index series of species of graphs and networs A species is a class C of labelled graphs or networs which is closed under isomorphism. The underlying set of a graph is its vertex set, the underlying set of a networ is the set of its internal vertices, and any isomorphism is induced by a relabelling along a bijection between the underlying sets. A species C is said to be weighted if each structure s of C is assigned a weight (ii)

4 w(s) taen from a commutative ring such that the weight function s w(s) is invariant under isomorphism. For example, given a graph G, we can define the weight w 0 (G) = y m, where m = E(G) and y is a formal variable acting as an edge counter. A species weighted with a function w are denoted by C w. For a species G of graphs weighted with w 0 (G) = y m, the exponential generating function is G(x,y) := G w0 (x) = g n (y) xn n!, n 0 where g n (y) = m 0 g n,my m, g n,m is the number of graphs in G over the set [n] of vertices and having m edges; the ordinary generating function is G(x,y) := G w0 (x) = g n (y)x n, n 0 where g n (y) = m 0 g n,my m, g n,m is the number of isomorphism classes of graphs in G having n vertices and m edges. Analogous generating functions are defined for a species N of -pole networs having n internal vertices, m edges and weighted with the function w 0 (N) = y m. Note that unlabelled - pole networs are isomorphism classes of networs, where any isomorphism ϕ : N N is assumed to be pole-preserving, i.e. ϕ(0) = 0 and ϕ() =. In particular, any automorphism of a networ N should be pole-preserving. We also need to consider the subclass N τ of N consisting of τ-symmetric networs, i.e. networs N such that τ N is isomorphic to N. Suppose we are given a species G of graphs and a symmetric species N of networs such that the composition G N is canonical. In [3] we prove, following Walsh [6], that (G N)(x,y) = G(x, N(x,y)) for the labelled graphs of G N. Here we rigorize and extend the method of Walsh [7] to compute the tilde generating function (G N) (x, y) for unlabelled (G N)-structures. The method of Walsh involves special cycle index series W G (a;b;c), W + N (a;b;c) and W N (a;b;c) in variables a = (a,a,...), b = (b,b,...) and c = (c,c,...). These series are defined as follows. Let G = (V (G),E(G)) be a graph in G. A permutation σ of V (G), which is an automorphism of the graph G, induces a permutation σ () of the set E(G) of edges whose cycles are of two possible sorts: if c is a cycle of σ () of length l, then either σ l (a) = a and σ l (b) = b for each edge e = ab of c (a cylindrical edge cycle), or else σ l (a) = b and σ l (b) = a for each edge e = ab of c (a Möbius edge cycle). For example, the automorphism σ = (,, 3, 4)(5, 6, 7, 8) of the graph of Figure 3(i) induces the cylindrical edge cycle (5, 6, 37, 48), and the automorphism σ = (,, 3, 4, 5, 6, 7, 8) of the graph of Figure 3(ii) induces the Möbius edge cycle (5, 6, 37, 48).

5 (i) (ii) Fig. 3. (i) Cylindrical edge cycle, (ii) Möbius edge cycle. For an automorphism σ Aut(G) of G, denote by σ the number of cycles of length of σ, by cyl (G,σ) the number of cylindrical edge cycles of length, and by möb (G,σ) the number of Möbius edge cycles of length induced by σ in G. Given a graph G G and an automorphism σ of G, the weight w(g,σ) of such a structure is the following cycle index monomial: w(g,σ) = a σ a σ b cyl (G,σ) b cyl (G,σ) c möb (G,σ) c möb (G,σ). Then the Walsh index series W G (a;b;c) of a species G of graphs is defined as W G (a;b;c) = G Typ(G) Aut(G) σ Aut(G) w(g,σ), where the notation G Typ(G) means that the summation should be taen over a set of representatives G of the isomorphism classes of graphs in G. Let σ S[U] be a permutation of the underlying set U of a networ N. We can extend σ to σ + = (0)()σ and to σ = (0, )σ to account for the poles. Denote by ˆN the graph on U {0, } corresponding to a networ N N. Then Aut + (N) = {σ S[U] σ + Aut( ˆN)} and Aut (N) = {σ S[U] σ Aut( ˆN)}. Notice that Aut + (N) = Aut(N). For N N and σ Aut + (N), we assign the weight w(n,σ) = w( ˆN,σ + ), and for N N and σ Aut (N), a we set w(n,σ) = w( ˆN,σ ) a. In other words, only the internal vertex cycles are accounted for. Then, for a species N of networs, the Walsh index series are W + N (a;b;c) = W N (a;b;c) = N Typ(N) N Typ(N τ) Aut + (N) Aut (N) σ Aut + (N) σ Aut (N) w(n,σ), w(n,σ). Proposition 3. Let G be a species of graphs and N be a species of networs. Then the following series identities hold: and

6 G(x,y) =W G (x,x,x 3,...;y,y,y 3,...;y,y,y 3,...), Ñ(x,y) =W + N (x,x,x 3,...;y,y,y 3,...;y,y,y 3,...), Ñ τ (x,y) =W N (x,x,x 3,...;y,y,y 3,...;y,y,y 3,...). Denote by (W + N ) = (W + N ) (a;b;c) = W + N (a,a,...;b,b,...;c,c,...) and by (W N ) = (W N ) (a;b;c) = W N (a,a,...;b,b,...;c,c,...). Theorem 3. Let G be a species of graphs and N be a symmetric species of networs. Then the Walsh index series of the species G N is given by W G N (a;b;c) =W G (a,a,...;(w + N ), (W + N ),...;(W N ), (W N ),...), and the generating function (G N) (x,y) of unlabelled (G N)-structures is (G N) (x,y) = W G (x,x,...;ñ(x,y), Ñ(x,y ),...;Ñτ(x,y), Ñτ(x,y ),...). For a species N B of networs corresponding to a species B of -connected graphs (K B), we have W + N B (a;b;c) = ( + b ) a b W B (a;b;c) and W N B (a;b;c) = ( + c ) a c W B (a;b;c) (see [7]). 4 Walsh index series for the toroidal cores The Walsh index series for the cores K 5, M, and M can be computed by hand, and W + K 5 \e (a;b;c) and W K 5 \e (a;b;c) can be derived from W K 5 (a;b;c) (see [5]). For the Walsh index series W H of the class H of crowns of K 5 \enetwors, we use a variant of Theorem 3., where the K 5 \e-networs are substituted for some edges of a cycle C n (n 3), and the unsubstituted edges form a matching µ of C n. The homogeneous matching polynomial of G is M G (y,z) = y µ z m µ, µ M(G) where M(G) is the set of matchings of the graph G and m = E(G). In particular, we have the homogeneous matching polynomials U n (y,z) = M Pn (y,z) and T n (y,z) = M Cn (y,z), where P n is the path, and C n is the cycle over the set of vertices V = {,,...,n} (see [4]). They satisfy the recurrence relations U n (y,z) = yzu n (y,z) + zu n (y,z), T n (y,z) = yz U n (y,z) + zu n (y,z), for n 3, with the initial values U (y,z) =, U (y,z) = y +z, and T (y,z) = z, T (y,z) = yz + z. Given a graph G with a matching µ, the edges of G are partitioned into two sorts Y and Z depending if they are in µ (sort Y ) or not (sort Z). An and

7 automorphism σ of a matched graph (G,µ) is an automorphism σ of G that leaves the matching fixed, i.e. σ(µ) = µ. This induces cylindrical and Möbius edge cycles of sort Y, counted by the variables b,b,... and c,c,..., respectively, and cylindrical and Möbius edge cycles of sort Z, counted by the variables β,β,... and γ,γ,..., respectively. Let w µ (σ) denote the cycle index monomial: b cyl,y (σ) w µ (σ) = a σ γ möb,z(σ), where cyl,y (σ), möb,y (σ), cyl,z (σ) and möb,z (σ) denote the number of cylindrical and Möbius edge cycles of length and of sort Y or Z, respectively. Then we set W m G (a,b,c,β,γ) = G Typ(G) c möb,y (σ) Aut(G) β cyl,z (σ) σ Aut(G) µ Fix m (σ) w µ (σ), where Fix m (σ) denotes the set of matchings µ of G fixed by σ. Proposition 4. The Walsh index series W G m Z N(a;b;c) can be obtained from the extended Walsh index series WG m (a;b;c;β;γ) by W G m Z N(a;b;c) = W m G (a;b;c; (W + N ), (W + N ),...;(W N ), (W N ),...). By applying Proposition 4. with G = C = n 3 C n and N = K 5 \e, W H (a;b;c) for the species H of crowns of K 5 \e-networs can be calculated as W H (a;b;c) = W m C (a;b;c; (W + K 5 \e ), (W + K 5 \e ),...;(W K 5 \e ), (W K 5 \e ),...). The extended Walsh index series W m C n (a;b;c;β;γ) for matched cycles of size n, n 3, can be computed by using the extended Walsh index series of matched paths and the Walsh index series of cycles (see [5]). The result is Theorem 4. The extended Walsh index series WC m n, n 3, is given by WC m n (a;b;c;β;γ) = φ( n n d )ad nt d (b n,β n ) d d d for n odd, and by W m C n (a;b;c;β;γ) = n d n + a a n d n (β γ U n (b,β ) + βc U n 3(b,β )), φ( n d )ad nt d (b n,β n ) + d d d 4 [a a n β Un +a n (γun (b,β ) + c γ β U n (b,β ) +c β Un 4(b,β ))], for n even, where φ is the Euler φ-function. (b,β )

8 Thus, we can compute the Walsh index series W H (a;b;c) of the species H of crowns of K 5 \e-networs by using W m C = n 3 W m C n, W + K 5 \e, and W K 5 \e in Proposition 4.. Computational results are presented in Table. To count Table The number t C (n) of unlabelled toroidal cores (having n vertices). n t C (n) n t C (n) n t C (n) unlabelled toroidal graphs with no K 3,3 s, it is necessary to use the generating functions for unlabelled planar networs N P and unlabelled τ-symmetric planar networs N P,τ. It is still not nown how to compute efficiently these power series. However, we enumerated planar networs of small size (see [5]). References [] Bergeron, F., G. Labelle, and P. Leroux, Combinatorial Species and Tree-lie Structures, Cambrige Univ. Press, 998. [] Gagarin, A., and W. Kocay, Embedding graphs containing K 5 -subdivisions, Ars Combin. 64 (00), [3] Gagarin, A., G. Labelle, and P. Leroux, Structure and labelled enumeration of K 3,3 -subdivision-free projective-planar graphs, Pure Math. Appl. (005), to appear. Preprint: arxiv:math.co/ [4] Gagarin, A., G. Labelle, and P. Leroux, Characterization and enumeration of toroidal K 3,3 -subdivision-free graphs, submitted, 004. Preprint: arxiv:math.co/ [5] Gagarin, A., G. Labelle, and P. Leroux, Counting unlabelled toroidal graphs with no K 3,3 -subdivisions, Adv. in Appl. Math. (006), to appear. Preprint: arxiv:math.co/ [6] Walsh, T.R.S., Counting labelled three-connected and homeomorphically irreducible two-connected graphs, J. Combin. Theory Ser. B 3 (98),. [7] Walsh, T.R.S., Counting unlabelled three-connected and homeomorphically irreducible two-connected graphs, J. Combin. Theory Ser. B 3 (98), 3.

Forbidden minors and subdivisions for toroidal graphs with no K 3,3 s

Forbidden minors and subdivisions for toroidal graphs with no K 3,3 s Andrei Gagarin 2 Laboratoire de Combinatoire et d Informatique Mathématique (LaCIM) Université du Québec à Montréal (UQAM) Montréal,