and Timothy Walsh. 1 ... B = BF denotes the class of 2-connected graphs all of whose. 3-connected components are in F. Note: K2 is in B. We introduce the ...
2-connected graphs with prescribed three-connected components by Pierre Leroux, LaCIM et D´epartement de math´ematiques, Universit´e du Qu´ebec `a Montr´eal Ottawa-Carleton Discrete Mathematics Workshop Carleton University, Ottawa, May 25–26, 2007 Work in collaboration with Andrei Gagarin, Gilbert Labelle and Timothy Walsh
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Introduction Our goal : 1. An analogue of the block-cutpoint tree bc(g) of a connected graph g, for 2-connected graphs 2. Functional equations for 2-connected graphs and networks with prescribed 3-connected components 3. Enumerative consequences and applications
Figure 1: A 2-connected graph
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Figure 2: A separating pair
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Figure 3: Bond along a separating pair 3
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Figure 4: Separating pairs
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Figure 5: Separating pairs and tricomponents Tricomponents: 3-connected graphs (A, B, and C) or polygons: (S, T, U, and P ).
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Dissymmetry Theorem for 2-connected graphs Let F be a given class of 3-connected graphs. B = BF denotes the class of 2-connected graphs all of whose 3-connected components are in F . Note: K2 is in B. We introduce the following classes of rooted graphs in B: B ◦ : with a distinguished tricomponent; B • : with a distinguished bond; B ◦−• : with a distinguished pair of adjacent tricomponent and bond. Theorem. We have the following identity (species isomophism): B ◦ + B • = B − K2 + B ◦−• . Proof. Analyse the relative positions of the rooted tricomponents or bonds with respect to the graph ”centers”.
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2-pole networks A 2-pole network (or simply a network ) is a connected graph N with two distinguished vertices 0 and 1, such that the graph N ∪ 01 is unseparable. The vertices 0 and 1 are called the poles of N , and all the other vertices of N are said to be internal. The internal vertices of a network form its underlying set 1 x
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Figure 7: (i) series-parallel network (ii) (K4 )0,1 - network
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Figure 8: The trivial networks: (i) 11 (ii) y 11
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For any class of graphs G, we define an associated class of networks G0,1 . A network in G0,1 is obtained from a graph in G by selecting and removing an edge and relabelling its endpoints with 0 and 1. 1
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Figure 9: A (K5 )0,1 - network For memory, the corresponding exponential generating functions satisfy ∂ 2 x G0,1 (x, y) = 2 G(x, y) ∂y We define an operator τ acting on 2-pole networks, N 7−→ τ ·N , which interchanges the poles 0 and 1. A class N of networks is called symmetric if N ∈ N =⇒ τ · N ∈ N .
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The composition M ↑ N Let M be a class of graphs (or networks) and N be a symmetric class of networks. We denote by M ↑ N the class of pairs of graphs (or networks) (M, T ), such that 1. the graph (or network) M (called the core) is in M, 2. the vertex set V (M ) is a subset of V (T ), 3. there exists a family {Ne } of networks in N (called the components) such that the graph T can be obtained from M by substituting Ne for each edge e of M , the poles of Ne being identified with the extremities of e. 1
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Figure 10: Example of a (M ↑ N )-structure (M, T )
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Examples of compositions 1. If we take the class G = {K2 } for cores and the class N of all networks for components, then the (K2 ↑ N )-structures consist of graphs G together with two selected (adjacent or not) vertices a and b, such that the graph G ∪ ab is 2-connected. 2. The composition M ↑ N is called canonical if for any structure (M, T ) ∈ M ↑ N , the core M ∈ M is uniquely determined by the graph (or network) T . In this case, we can identify M ↑ N with the class of resulting networks T . For example, we can take G = K, the class of complete graphs, N = 11 + y 11, the class of trivial networks, (see Figure 4). Then we have K ↑ (11 + y 11) = G , where G denotes the class of all graphs, the composition being canonical.
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Let F be a given class of 3-connected graphs. B = BF denotes the class of 2-connected graphs all of whose 3-connected components are in F . D = DF denotes the class of networks all of whose 3-connected components are in F . In fact D = (1 + y)B0,1 − 11. Theorem. We have the following identities: B ◦ = F ↑ D + C ↑ (D − S) B • = K2 ↑ (1 + y)E≥2 (H + S) − E2 (S) ◦−• 2 B = K2 ↑ (1 + y)(H + S)E≥1 (H + S) − S where C is the class of unoriented cycles of length ≥ 3 (polygons), S is the class of (strictly) series networks, H = F0,1 ↑ D (irreducible networks).
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Theorem. (Trakhtenbrot) The species of two-pole networks D corresponding to a species of 3-connected graphs F can be expressed by the relation D = (1 + y)E(F0,1
XD 2 ↑D+ ) − 11. 1 + XD
Proof. It follows from the tricomponent decomposition of a 2-connected graph that D = y 11 + (1 + y)(H + S + E≥2 (H + S)) = (1 + y)E(H + S) − 11 , where H = F0,1 ↑ D and S satisfies S = (D − S) ·s D = (D − S)XD and hence
XD 2 . S= 1 + XD
Corollary. In terms of the exponential generating functions we have 2 xD (x, y) D(x, y) = (1 + y) exp F0,1 (x, D(x, y)) + − 1. 1 + xD(x, y) 12
References [1] E.A. Bender, Zh. Gao, and N.C. Wormald, “The number of labeled 2-connected planar graphs”, Electron. J. Combin. 9 (2002), Research Paper 43, 13 pp. [2] A. Gagarin, G. Labelle and P. Leroux, “Counting unlabelled toroidal graphs with no K33 -subdivisions”, Advances in Applied Mathematics, 39 (2007), 51–75. [3] J. E. Hopcroft and R. E. Tarjan, “Dividing a graph into triconnected components”, SIAM J. Comput. 2 1973, 135– 158. [4] W. T. Tutte, Graph Theory. Encyclopedia of Mathematics, vol. 21, 1984. [5] T.R. Walsh, “Counting labelled three-connected and homeomorphically irreducible two-connected graphs”, J. Comb. Theory Ser. B 32 (1982), 1–11. [6] T.R. Walsh, “Counting unlabelled three-connected and homeomorphically irreducible two-connected graphs”, J. Comb. Theory Ser. B 32 (1982), 12–32. 13
