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Graph representations of a bicircular matroid
Discrete Applied Mathematics 118 (2002) 249–262
Graph representations of a bicircular matroid Nancy Ann Neudauer Department of Mathematics, Pacic Lutheran University, Tacoma, WA 98447, USA
Received 14 August 1998; received in revised form 9 October 2000; accepted 22 January 2001
Abstract The bicircular matroid B(G) of a graph G is known to be a transversal matroid. There are, in general, many graphs that represent the matroid as well as many presentations. We discuss the graphs that represent the same bicircular matroid. Given any presentation of a bicircular matroid, we show how to .nd a graph representing the matroid, and that, in some cases, there is more than one such graph. In the .rst four sections, we describe background and pertinent results on bicircular matroids. Many of the lemmas and theorems in these sections have straightforward proofs but these results have not been previously stated. In the .nal section, we illustrate how the graph constructed via the techniques developed by Brualdi and Neudauer (Quart. J. Math. Oxford (2) 48 (1997) 17) for .nding the minimal presentations of a bicircular matroid, combined with the earlier results of this paper, relate to the operations developed by Coullard et al. (Discrete Appl. Math. 32 (1991) 223). ? 2002 Elsevier Science B.V. All rights reserved.
1. Introduction Let G be a graph (loops and parallel edges allowed) with vertex set V = {1; 2; : : : ; n} and edge set E. The bicircular matroid of G is the matroid 1 B(G) de.ned on E whose circuits are the subgraphs which are subdivisions of one of the graphs: (i) two loops on the same vertex, (ii) two loops joined by an edge, (iii) three edges joining the same pair of vertices. Here and elsewhere, we identify a subgraph and its collection of edges. The circuits of B(G) are the bicycles of G (Fig. 1). A set of edges is independent in B(G) provided that each connected component contains at most one cycle of G. The (bicircular matroid) rank of a set X of edges is (X ) = n(X ) − t(X ), where n(X ) is the number of vertices incident with the edges of X and t(X ) is the number of (non-trivial) tree components of X . If G is a tree, then E is an independent set and
E-mail address: [email protected] (N.A. Neudauer). 1 A general reference for matroid theory is [8]. 0166-218X/02/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 6 - 2 1 8 X ( 0 1 ) 0 0 2 1 0 - 4
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Fig. 1. Bicycles of G.
hence a basis of B(G). If G is a connected graph that is not a tree, then the bases of B(G) are the spanning subgraphs of G each of whose connected components is a unicyclic subgraph of G; in particular the bases have cardinality equal to n. Let A = (A1 ; A2 ; : : : ; An ) be an arbitrary family of subsets of a set E. A subset T of E is a transversal of (A1 ; A2 ; : : : ; An ) if there is a bijection : T → {1; 2; : : : ; n} such that x ∈ A (x) for all x ∈ T . X is a partial transversal of (A1 ; A2 ; : : : ; An ) if there exists an injective map : X → {1; 2; : : : ; n} such that x ∈ A(x) for all x ∈ X . The set of partial transversals of A are the independent sets of a matroid on E (see e.g. [8]). The matroid is denoted M (A). If M is an arbitrary matroid and M = M (A), we call M a transversal matroid and A a presentation of M. As shown by Matthews [7] bicircular matroids are transversal matroids. Let Ei be the set of edges of G which are incident with vertex i (i = 1; 2; : : : ; n). Then B(G) is the transversal matroid whose independent sets are the partial transversals of the family of sets E(G) = (E1 ; E2 ; : : : ; En ). The family of sets E(G) is the natural presentation of B(G) corresponding to the graph G. Let A = (A1 ; A2 ; : : : ; An ) be an arbitrary family of subsets of a set E. If M (A) has rank k and (Ai1 ; Ai2 ; : : : ; Aik ) (1 6 i1 ¡ · · · ¡ ik 6 n) has a transversal, then M (A) = M ((Ai1 ; Ai2 ; : : : ; Aik )) [5]. For this reason, we generally assume that (A1 ; A2 ; : : : ; An ) has a transversal. That is, M (A) has rank n. In general, there are many families B = (B1 ; B2 ; : : : ; Bn ) of subsets of E such that M = M (B). It is well-known (shown in [2]) that there exists a unique maximal presentation 2 (M1 ; M2 ; : : : ; Mn ) of M. Here maximal means that for each i and for each x ∈ Mi , (M1 ; : : : ; Mi−1 ; Mi ∪ {x}; Mi+1 ; : : : ; Mn ) is not a presentation of M. The transversal matroid M has, in general, many minimal presentations (C1 ; C2 ; : : : ; Cn ). The sets Ci in each minimal presentation are distinct cocircuits of M. Conversely, a presentation consisting of cocircuits is minimal. Note that if e is a coloop of M, then {e} is a cocircuit and is a set in every minimal presentation of M. Two presentations A = (A1 ; A2 ; : : : ; An ) and B = (B1 ; B2 ; : : : ; Bn ) are isomorphic, written A ∼ = B, if there exists a bijection : {1; 2; : : : ; n} → {1; 2; : : : ; n} such that Ai = B(i) for all i. We say that a presentation A = (A1 ; A2 ; : : : ; An ) is contained in a presentation B = (B1 ; B2 ; : : : ; Bn ) if there exists a bijection : {1; 2; : : : ; n} → {1; 2; : : : ; n} such that Ai ⊆ B(i) for all i ∈ {1; 2; : : : ; n}. 2
Apart from the order of the sets.
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In this paper, we discuss the graphs that give rise to the same bicircular matroid. We give a characterization of bicircular matroids in terms of the presentations. We then show how, given any presentation of a bicircular matroid, to .nd a graph representing the matroid, and that, in some cases, there is more than one such graph. Once we .nd the graph, by the results of Brualdi and Neudauer [4] we can .nd the other minimal presentations, and then other graphs representing this bicircular matroid. We show that a presentation with the property that the intersection of any three sets is empty is the natural presentation for some graph G that represents B(G). By .nding all such presentations, we show that we can .nd all graphs representing the bicircular matroid. In general, we consider the various graphs that can represent one bicircular matroid, and how those graphs are related to each other. In the .nal section, we show that if we use the methods in [4] to .nd the minimal presentations of B(G) that have empty intersection of any three sets, the graphs G that arise from these new presentations are related to G by the operations described in [6].
2. Basic properties Let G be a graph with vertex set V = {1; 2; : : : ; n} and edge set E. Since the bicircular matroid B(G) is the direct sum of the bicircular matroids of its connected components, we assume that G is connected. If G has a pendant edge e, then e is in every basis of B(G) and hence is a coloop of B(G). Henceforth, we assume that every vertex of G is incident with at least two edges, that is, the minimum degree of G is at least 2. We .rst identify the cocircuits of the bicircular matroid B(G) (proved in [4]). Lemma 1. A subset C of the edges E of G is a cocircuit of B(G) if and only if the spanning subgraph G\C obtained from G by deleting the edges in C has exactly one tree component T and each edge in C is incident with at least one vertex of T . The following lemma is proved in [7]. Lemma 2. Let G be a connected graph with at least n + 1 edges. Then B(G) is a connected matroid if and only if G has no pendant edges. In particular; if G has no pendant edges; then each edge of G is in a bicycle. A graph is called bicyclic provided each edge is in a bicycle. Thus a graph is bicyclic if and only if its bicircular matroid has no coloops. Given any graph G, we can easily .nd a presentation of B(G), the natural presentation E(G). The natural presentation has the property that the intersection of any three sets is empty because no edge is adjacent to three vertices of the graph. Each edge is in at most two sets of the natural presentation (one only if the edge is a loop). A presentation with the property that the intersection of any three sets is empty is said to have the triple intersection property. Given an arbitrary presentation of a
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Fig. 2. Wheel on six vertices.
transversal matroid, M, we can determine if M is a bicircular matroid using the following characterization of Matthews [7]. Theorem 3. Bicircular matroids are precisely the loopless transversal matroids which have a presentation such that the intersection of any three members is empty. A natural question that arises is whether every minimal presentation of a bicircular matroid has the triple intersection property. We can see with the example of a wheel centered at vertex 1, given in Fig. 2, that this is not true. Using the results of Brualdi and Neudauer in [4], all the minimal presentations for the graph can be determined. In the case of the wheel in Fig. 2, the natural presentation is a minimal presentation, and the natural presentation has the triple intersection property. We can .nd another minimal presentation of B(W ) by replacing E1 with a set of edges that is the complement of a spanning tree of W [4]. So, another minimal presentation for B(W ) is A(W ) = (C1 ; E2 ; E3 ; E4 ; E5 ; E6 ; E7 ), where C1 = {a; b; f; j; k; l}, E2 = {a; g; h}, E3 = {b; h; i}; E4 = {c; i; j}; E5 = {d; j; k}; E6 = {e; k; l}; E7 = {f; l; g}. This presentation has three sets with non-empty intersection. We see that it is possible to have one minimal presentation with the triple intersection property and another without this property. We know by Theorem 3 that at least one presentation of a bicircular matroid has the triple intersection property. If this presentation is not minimal, it must contain a minimal presentation, and this minimal presentation also has this property. Thus, at least one of the minimal presentations of a bicircular matroid will have the triple intersection property. If M is a bicircular matroid and G is a graph such that M = B(G) then G is a representation of M, and we say that G represents M. If G is a graph with minimum degree at least three and parallel edges allowed there can be more than one graph representing B(G). In the following example, the graphs G in Fig. 3 and G in Fig. 4 represent the same bicircular matroid. The natural presentation E(G) = (E1 ; E2 ; E3 ; E4 ), where E1 = {a; b; c}; E2 = {a; b; d}; E3 = {c; d; e; f; g}, E4 = {e; f; g} is not a minimal presentation for the matroid.
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Fig. 3. G represents B(G).
Fig. 4. G also represents B(G).
As shown in [4], we can .nd a minimal presentation for B(G) by removing from E3 one edge from vertex 3 to the tree component of G3 , which is vertex 4. This gives us a minimal presentation with the triple intersection property, which is the natural presentation with respect to G . Here E(G ) = (E1 ; E2 ; E3 ; E4 ), where E1 = {a; b; c}, E2 = {a; b; d}, E3 = {c; d; e; g}, E4 = {e; f; g}. Lemma 4. Let G be a graph; and let E(G) be the natural presentation of the bicircular matroid of G. Then G is determined up to isomorphism by E(G). Proof. Let G and H be two arbitrary graphs with vertices {1; 2; : : : ; n} such that G = H . Since they are not the same graph, there is some edge e incident with vertex v in G that is not incident with vertex v in H . In a natural presentation, ej ∈ Ej if and only if ej is incident with vertex j. So e ∈ Ev in the natural presentation of B(G), but e ∈ Ev in the natural presentation of B(H ), and the natural presentations are not the same. So the map from the natural presentation to the graph is well de.ned. Let G be a graph with natural presentation E(G) = (E1 ; E2 ; : : : ; En ), and H be a graph with natural presentation E(H ) = (Ei1 ; Ei2 ; : : : ; Ein ). Assume G ∼ = H . Then there are bijections : V (G) → V (H ) and % : E(G) → E(H ) such that the edge e of G is incident with a vertex v of G if and only if %(e) is incident with (v). Then : j → ij and e ∈ Ej if and only if %(e) ∈ E ( j) , and E(G) ∼ = E(H ). Assume E(G) ∼ = E(H ). Then there exist bijections & : E(G) → E(H ) and : j → ij such that &(Ej ) = E( j) and e ∈ Ej if and only if &(e) ∈ E( j) . Then e is incident with vertex j of G if and only if &(e) is incident with (j), and G ∼ = H.
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Hereafter if we say G = H for two graphs G and H , we mean up to relabelling of the vertices and the edges. The following theorem follows from [4] and from Lemma 4. Theorem 5. Let G be a graph with no parallel edges and minimum degree at least three. If G is a graph such that B(G) = B(G ); then G ∼ = G . Proof. If G is such a graph, the natural presentation is a minimal presentation. The only set in any presentation that diLers from a set of the natural presentation is Ei , where i is the center of a wheel [4]. In this case, we obtain another minimal presentation by replacing the edges of Ei that are spokes of the wheel with the complement of a spanning tree of the wheel. Then any such cocircuit Ci that diLers from Ei has in it some edge from the rim of the wheel. Each rim edge is necessarily in at least two sets of any presentation. For, if {j; k} is a rim edge, it is in Ej and Ek . G\Ej has no coloops in the matroid B(G\Ej ) and G\Ek has no coloops in the matroid B(G\Ek ), so Ej and Ek are sets in every presentation of B(G). If {j; k} is also in Ci , it is in three sets of this minimal presentation. Thus no other presentation has the triple intersection property. If there were another graph, G , representing B(G), there would be a presentation of B(G) with the triple intersection property which is the natural presentation of B(G) with respect to G . By Lemma 4, the natural presentation determines the graph up to isomorphism. So, G ∼ = G . So, if G has no parallel edges and minimal degree at least three, there is a unique graph, up to isomorphism, representing B(G). If G is a graph with minimum degree at least three and parallel edges allowed, there can be more than one graph representing B(G). Coullard et al. [6] characterize those graphs that represent the same bicircular matroid, and we discuss their work in Section 4.
3. Construction of the graphs from a presentation Given an arbitrary presentation of a transversal matroid, M, we can determine if M is a bicircular matroid: from an arbitrary presentation we can .nd the maximal presentation by an algorithm of Brualdi and Dinolt [3], and from this we can .nd the other presentations of the matroid by Bondy and Welsh [1]. If there is a presentation with the triple intersection property, we can .nd it using these results. This allows us to determine, although not eMciently, whether a transversal matroid is a bicircular matroid. If M is a bicircular matroid, we can .nd a graph representing M. In general, there is not a unique graph representing the matroid. Algorithm 6. Given any presentation for a bicircular matroid M the graphs G representing M such that B(G) = M can be determined as follows: (1) Determine all the presentations of M (see [1]).
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(2) For each presentation A with the intersection of any three sets empty; construct a graph G from A by putting |Ai ∩ Aj | edges between vertex i and vertex j. If an edge appears in only one set; put a loop at that vertex. Notice that the presentation, A, with the triple intersection property is the natural presentation for B(G) with respect to G. Theorem 7. Given any presentation for a bicircular matroid M; then; up to isomorphism; Algorithm 6 determines all graphs G representing M such that B(G) = M. Proof. Suppose G represents M and E(G) is a natural presentation of M. By Lemma 4, a natural presentation determines the graph, up to isomorphism. A natural presentation always has the triple intersection property. So the algorithm produces all graphs representing M. Coullard et al. [6] showed Lemma 8. Let G be a graph such that B(G) is connected. Then there exists a graph representation H of B(G) such that the star of every vertex of H is a cocircuit of B(G). Here the star of a vertex is the set of edges incident with that vertex. Corollary 9. Let M be a bicircular matroid and let A be a minimal presentation of M with the triple intersection property. Then there exists a graph representation H of M such that every cocircuit of A is the star of a vertex of H . Proof. A is a minimal presentation of M such that the intersection of any three sets of A is empty. Using Algorithm 6 we can construct a graph representing M with A as its natural presentation. Each cocircuit, Ai , of A is composed of the edges of the graph incident with vertex i. So the set of edges Ai is the star of vertex i. From a presentation of a bicircular matroid we can thus .nd the collection of graphs representing a bicircular matroid. In the next section we describe how Coullard et al. [6] characterized the collection of graphs having the same bicircular matroid as that of a given graph. 4. Graphs that represent the same bicircular matroid A natural question that arises from the preceding discussion is how graphs G and G are related if B(G) is also represented by G . In this section we will discuss some results of Coullard et al. [6] in which they characterize the collection of graphs that represent a given bicircular matroid.
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Fig. 5. Balloons of G.
We begin with some notation and de.nitions. In [9] Wagner discusses matroid connectivity further. A cut vertex of a connected graph is one whose deletion produces a disconnected graph. Let F be a proper subset of E, the edges of the graph G, and let H be the subgraph of G induced by F. The set of vertices common to H and to G[E −E(H )] are the vertices of attachment of H . A block of G is a maximal subgraph satisfying the property that every pair of edges is contained in a cycle. An end-block of G is a block H having exactly one vertex of attachment, called the tip of H . Coullard et al. [6] de.ne a balloon of G as a maximal set B of edges such that the subgraph induced by B is a subdivision of one of the graphs in Fig. 5, and the only vertex of attachment of the balloon is v as indicated in the .gure. A line is a set of edges not contained in any balloon, that forms a path, the internal vertices of which have degree two in G and the end vertices of which have degree at least three. Following is a description of operations from [6] which, when applied to a graph, produce a graph with the same bicircular matroid. Let S be a line of G having end vertices u and v, and let e be the unique edge of S incident to v. Rede.ne the incidence relation of e so that e is incident to a vertex w = v of S instead of v. This new graph, G is obtained from G by a rolling of S away from v, and G is obtained from G by an unrolling of S to v. S is a balloon of G . The following two theorems were .rst proved by Wagner [9] for the case where B(G) is 3-connected, and later by Coullard et al. [6] when B(G) is connected. Theorem 10. Let G and G be graphs such that B(G) is connected and G is obtained from G by a rolling of a line S away from one of its end vertices v. Then B(G) = B(G ) if and only if there exists an end-block H of G such that S ⊆ E(H ); v is a tip of H; and every cycle of H contains v. Let v be a vertex incident to exactly three lines L1 , L2 , and L3 in G. Suppose the other end vertex of L1 is u, and the other end of L2 and L3 is w = u. Let e1 be the edge of L1 incident to u, and let e2 be the edge of L2 incident to w. Rede.ne the incidence relations of e1 and e2 so that e1 is incident to w instead of u, and e2 is incident to u instead of w. This new graph, G is obtained from G by a rotation of L1 and L2 at the vertex v. Theorem 11. With the above notation; let G and G be graphs such that B(G) is connected and G is obtained from G by a rotation of lines L1 and L2 at v; an end
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vertex incident with both L1 and L2 . Then B(G) = B(G ) if and only if there exists 3 an endblock H of G such that L:= i=1 Li is contained in E(H ); u is a tip of H and every cycle of H \L contains u; where L3 is the line having the same ends as L2 and u is the end of L1 not equal to v. Let S be a line (respectively, balloon) of G. Let G be a graph obtained from G by replacing S with another line (respectively, balloon) on the same edge set and having the same vertices (respectively, vertex) of attachment. Then G is obtained from G by replacement. Also, if G and G are graphs on the same edge set and each graph is a bicycle, then G is obtained from G by replacement. Theorem 12. Let G and G be graphs such that G is obtained from G by replacement. Then B(G) = B(G ). Rollings, unrollings, rotations, and replacements are operations. If G is obtained from G by an operation and B(G) = B(G ), we say that the operation is legitimate, and that G is r-equivalent to G. We de.ne two graphs to be b-equivalent if they have the same bicircular matroid. Thus r-equivalence implies b-equivalence. The converse is true if we rule out some small cases. Coullard et al. [6] proved. Theorem 13. Let G and G be b-equivalent graphs such that B(G) is connected and |V (G)| = |V (G )| ¿ 5. Then G and G are r-equivalent.
5. Graphs that arise from the minimal presentations of a bicircular matroid Brualdi and Neudauer [4] characterized the minimal presentations of a bicircular matroid B(G) in terms of the graph G. Earlier in this paper we showed that a presentation with the triple intersection property is the natural presentation for some graph G that represents B(G). Here we show that if we use the methods in [4] to .nd the minimal presentations of B(G) with the triple intersection property, the graphs G that have these new presentations as natural presentations are related to G by the operations developed by Coullard et al. [6] and described in Section 4. We want to show that every minimal presentation with the triple intersection property is the natural presentation for a graph that is related to our original graph by a series of legitimate operations described in Section 4. Throughout this section, G is a connected graph having minimum degree at least two and having at least .ve vertices. Lemma 14. Let G be a graph such that the natural presentation of B(G) is not minimal. Then there is a minimal presentation of B(G) contained in the natural presentation E(G). The graph that arises from this minimal presentation is obtained from G by either a series of replacements of balloons or rolling away of lines; and is r-equivalent to G.
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Proof. The natural presentation is not minimal if and only if there is a set Ei in the natural presentation whose removal leaves at least two trees by Lemma 1. Suppose G \ Ei has exactly two acyclic components for some i. One acyclic component is i, and i is the tip of an endblock containing the other tree. Every cycle in this endblock passes through i. Since Ei is not a cocircuit, by the results of [4], there is an edge {i; j} such that Ei \ {i; j} is a cocircuit. This edge was either in a balloon of G with vertex of attachment i, or it was in a line of G that has as one end vertex i. We have a new presentation with Ei replaced by Ci = Ei \{i; j}, and this presentation has the triple intersection property. The edge we removed from Ei now appears in only one set of the presentation, Ej . We construct a graph from this presentation by putting |Ck ∩ Cl | edges between vertex k and l, for each pair of vertices. There is a loop at vertex j. This new graph, G , is obtained from G either by the replacement of a balloon, if {i; j} was in a balloon in G, or by the rolling away of a line from vertex i, if {i; j} was in a line in G. Thus, by Theorem 10, G is obtained from G by a legitimate operation, and G is r-equivalent to G. We repeat this for each acyclic component of G \ Ei other than {i} and for each vertex i for which Ei is not a cocircuit. In this way we construct a graph H that is obtained from G by a series of replacements of balloons and rolling away of lines, and is r-equivalent to G. We have B(G) = B(H ), and the natural presentation with respect to H is a minimal presentation for B(G). We now have a graph H that represents B(G) whose natural presentation, E(H ), is a minimal presentation. We now show that there may be a minimal presentation for B(G) that is diLerent from the natural presentation when G\Ei has a cycle component with (possibly trivial) trees attached. We de.ne a wheel with possibly multiple edge spokes to be a pinwheel. Notice that a wheel is a pinwheel, but not conversely. Lemma 15. Let G be a graph with minimum degree at least three; and let i be a vertex of G such that G\Ei has a component that is a cycle. Then there is a minimal presentation of B(G) with the triple intersection property such that the graph; G ; that arises from this presentation is obtained from G by rotation or by a series of unrolling and rolling away of lines; and G is r-equivalent to G. Proof. If G is a graph with minimum degree at least three and no parallel edges, then since there is a component of G\Ei that is a cycle, there is a wheel centered at vertex i. By Theorem 5, we know there is a unique 3 graph representing B(G), and none of the other presentations has the triple intersection property. Suppose G has minimum degree at least three with parallel edges. If G \ Ei has a cycle component, then either i is the center of a pinwheel or the cycle component consists of (i) two edges on two vertices or (ii) a single loop at some vertex j. 3
Up to isomorphism.
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If i is the center of a pinwheel, as in case of a wheel, which is shown in the proof of Theorem 5, the only presentation of this portion of the graph with the triple intersection property is the natural presentation. In any other presentation, we must use an edge from the rim of the pinwheel, and this edge is necessarily in two other sets of every presentation. If the cycle is two edges on two vertices, i is the tip of an endblock F. Notice 3 that this endblock, F, consists of several lines. L:= i=1 Li is contained in E(F), v is the vertex incident with all three lines, i is a tip of F and every cycle of F \ L contains i, where L3 is the line having the same ends as L2 and i is the end of L1 not equal to v. The other minimal presentations are composed of distinct cocircuits of the matroid. If we remove an edge e from Ei , we must replace it with an edge from the cycle [4]. We must also remove the edge we have put into Ei from one of the sets so that it is not in three sets of the presentation, and we must replace this with e. We can do this only if edge e is a coloop of the matroid B(G \ Ej ), where j is the other end vertex of lines L2 and L3 . Now we have a presentation that is made up of cocircuits with the triple intersection property. We have exchanged two edges between two sets of the presentation. We construct the graph from this new minimal presentation, and see that it is obtained from G by a rotation of L1 with one of the other lines at v. By Theorem 11 this is a legitimate operation, and the new graph is r-equivalent to G. If the cycle is the single loop at j, we can replace an edge {i; j} in Ei with the loop edge. Now edge {i; j} appears in only one set of the presentation, Ej , and the loop edge appears in Ei and Ej . When we construct the new graph from this presentation, these two edges have switched roles. This new graph is obtained from G by the unrolling of one line (the loop) and then the rolling away of another line (edge {i; j}). Since i is the tip of an endblock and every cycle of that endblock contains i, by Theorem 10 this is a legitimate operation, and the new graph is r-equivalent to G. Note that in this case we can add the loop edge to Ei without removing the edge {i; j}. In this case, the new graph is obtained by the unrolling of a line (the loop), and is also r-equivalent to G. Lemma 16. Let G be a graph for which the natural presentation of B(G) is a minimal presentation; and let i be a vertex of degree two. There is another minimal presentation of B(G) with the triple intersection property that replaces the set of the natural presentation Ei with another cocircuit. The graph that arises from this presentation is obtained from G by the replacement of a line by a line or the replacement of a balloon by a balloon; and is r-equivalent to G. Proof. Because the natural presentation of B(G) is a minimal presentation, all degree two vertices are either contained in a line or in the part of a balloon that is not a cycle. The cycle portion of a balloon is a single edge loop. The other minimal presentations of B(G) are obtained by letting the edges of the line or balloon be the vertices of a tree, and each pair of connected vertices is a cocircuit
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of the minimal presentation [4]. Since the presentation must have the triple intersection property, this tree has maximum degree two, and so is a path. Let u be a vertex of degree at least three that is adjacent to a degree two vertex and is not at the loop of a balloon. Every edge of the line or balloon is a coloop of G \ Eu . We can replace the edge of the line or balloon that is in Eu with any other edge of the line or balloon. To ensure that the presentation has the triple intersection property, we put in Eu an edge that is one of the pendant vertices of the tree above. We construct the graph from this new presentation. Every edge that was in a line is still in a line, and every edge that was in a balloon is still in a balloon, with the same vertices or vertex of attachment. The new graph is obtained from G by the replacement of a line by a line, or of a balloon by a balloon. By Theorem 12 this is a legitimate operation, and thus the new graph is r-equivalent to G. These three lemmas encompass the cases where the removal of a set of edges about a vertex yields some coloops, and thus there are other presentations of the matroid. This is all tied together in the following theorem. This theorem follows from Theorem 13. In fact, we could state and prove a stronger result here by removing the word “minimal”. We instead prove the following result because, as such, the proof of Theorem 17 and the preceding lemmas demonstrate the correlation between the techniques of Brualdi and Neudauer [4] and those of Coullard et al. [6]. Theorem 17. Let G be a connected graph having minimum degree at least two and having at least ve vertices. Every minimal presentation for B(G) with the triple intersection property is the natural presentation for a graph that is related to G by a series of legitimate operations; and this new graph is r-equivalent to G. Proof. If G is a graph with no parallel edges and minimum degree at least three, then up to isomorphism there is only one graph representing B(G) by Theorem 5. There is only one presentation of B(G) with the triple intersection property. If G is a graph with minimum degree at least three, for each vertex i, each of the components of Gi is one of the following three types: (x) a tree, (y) a cycle with a (possibly trivial) tree rooted at each vertex, and (z) a bicyclic graph with a (possibly trivial) tree rooted at each vertex. Case 1: i is a vertex of type (x). In this case the natural presentation is not minimal. Lemma 14 shows that we can .nd the minimal presentations with the triple intersection property. The graphs that arise from these presentations are obtained from G by either a series of replacements of balloons, or the rolling away of a lines, and are r-equivalent to G. Case 2: i is a vertex of type (y). While the tree rooted at each vertex does provide coloops of the matroid B(G \ Ei ), these edges do not replace any edges in Ei . The edges of these trees are already in two sets of the presentation. If j is a vertex of one
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of these trees, G \ Ej has no coloops in the matroid B(G \ Ej ). So the edges of those trees are always in the same two sets of the presentation, and cannot be added to Ei . By Lemma 15 we know that if G \ Ei has a component that is a cycle, there is another minimal presentation of B(G) with the triple intersection property, and the graph, G , that arises from this presentation is obtained from G by rotation or by a series of unrolling and rolling away of lines, and G is r-equivalent to G. Since the trees rooted at the vertices of the cycle do not provide any other minimal presentation with the triple intersection property, these are all the graphs we get in this case. Case 3: i is a vertex of type (z). We showed above that the trees rooted at each vertex do not provide additional minimal presentations with the triple intersection property. There are no other coloops of B(G \ Ei ). So, there are no graphs obtained from G by replacing Ei with another cocircuit, where i is a vertex of this type. If G is a graph with minimum degree at least two, there is a unique graph G ∗ , called the mothergraph of G, such that G ∗ has minimum degree at least three and G is a subdivision of G ∗ . Brualdi and Neudauer [4] characterized the minimal presentations of B(G) in terms of the minimal presentations of B(G ∗ ). By the above results we see how the graph constructed from the other minimal presentations of B(G ∗ ) is related to G ∗ , and then to G. We still need to consider the vertices of degree two. If the natural presentation for B(G) is not a minimal presentation, we can .nd a minimal presentation by Lemma 14. The graph, G , that arises from this presentation is r-equivalent to G. We can apply Lemma 16 to G , which also represents B(G). This lemma tells us that if i is a vertex of degree two and we .nd another minimal presentation of B(G) with the triple intersection property, the graph that arises from this presentation, G is obtained from G by the replacement of a line by a line, or the replacement of a balloon by a balloon, and is r-equivalent to G . Since G is r-equivalent to G, and G is r-equivalent to G , G is r-equivalent to G. Acknowledgements I am grateful to the referees for their detailed suggestions. References [1] J.A. Bondy, D.J.A. Welsh, Some results on transversal matroids and constructions for identically self-dual matroids, Quart. J. Math. Oxford (2) 22 (1971) 435–451. [2] R.A. Brualdi, Transversal matroids, in: N. White (Ed.), Combinatorial Geometries, Cambridge Univ. Press, Cambridge, 1985. [3] R.A. Brualdi, G.W. Dinolt, Characterizations of transversal matroids and their presentations, J. Combin. Theory 12 (1972) 268–286. [4] R.A. Brualdi, N.A. Neudauer, The minimal presentations of a bicircular matroid, Quart. J. Math. Oxford (2) 48 (1997) 17–26. [5] R.A. Brualdi, E.B. Scrimger, Exchange systems, matchings, and transversals, J. Combin. Theory 5 (1968) 244–257.
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[6] C.R. Coullard, J.G. del Greco, D.K. Wagner, Representations of bicircular matroids, Discrete Appl. Math. 32 (1991) 223–240. [7] L.R. Matthews, Bicircular matroids, Quart. J. Math. Oxford (2) 28 (1977) 213–228. [8] J. Oxley, Matroid Theory, Oxford University Press, Oxford, 1992. [9] D.K. Wagner, Connectivity in bicircular matroids, J. Combin. Theory Ser. B 39 (1985) 308–324.
Graph representations of a bicircular matroid Nancy Ann Neudauer Department of Mathematics, Pacic Lutheran University, Tacoma, WA 98447, USA
Received 14 August 1998; received in revised form 9 October 2000; accepted 22 January 2001
Abstract The bicircular matroid B(G) of a graph G is known to be a transversal matroid. There are, in general, many graphs that represent the matroid as well as many presentations. We discuss the graphs that represent the same bicircular matroid. Given any presentation of a bicircular matroid, we show how to .nd a graph representing the matroid, and that, in some cases, there is more than one such graph. In the .rst four sections, we describe background and pertinent results on bicircular matroids. Many of the lemmas and theorems in these sections have straightforward proofs but these results have not been previously stated. In the .nal section, we illustrate how the graph constructed via the techniques developed by Brualdi and Neudauer (Quart. J. Math. Oxford (2) 48 (1997) 17) for .nding the minimal presentations of a bicircular matroid, combined with the earlier results of this paper, relate to the operations developed by Coullard et al. (Discrete Appl. Math. 32 (1991) 223). ? 2002 Elsevier Science B.V. All rights reserved.
1. Introduction Let G be a graph (loops and parallel edges allowed) with vertex set V = {1; 2; : : : ; n} and edge set E. The bicircular matroid of G is the matroid 1 B(G) de.ned on E whose circuits are the subgraphs which are subdivisions of one of the graphs: (i) two loops on the same vertex, (ii) two loops joined by an edge, (iii) three edges joining the same pair of vertices. Here and elsewhere, we identify a subgraph and its collection of edges. The circuits of B(G) are the bicycles of G (Fig. 1). A set of edges is independent in B(G) provided that each connected component contains at most one cycle of G. The (bicircular matroid) rank of a set X of edges is (X ) = n(X ) − t(X ), where n(X ) is the number of vertices incident with the edges of X and t(X ) is the number of (non-trivial) tree components of X . If G is a tree, then E is an independent set and
E-mail address: [email protected] (N.A. Neudauer). 1 A general reference for matroid theory is [8]. 0166-218X/02/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 6 - 2 1 8 X ( 0 1 ) 0 0 2 1 0 - 4
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Fig. 1. Bicycles of G.
hence a basis of B(G). If G is a connected graph that is not a tree, then the bases of B(G) are the spanning subgraphs of G each of whose connected components is a unicyclic subgraph of G; in particular the bases have cardinality equal to n. Let A = (A1 ; A2 ; : : : ; An ) be an arbitrary family of subsets of a set E. A subset T of E is a transversal of (A1 ; A2 ; : : : ; An ) if there is a bijection : T → {1; 2; : : : ; n} such that x ∈ A (x) for all x ∈ T . X is a partial transversal of (A1 ; A2 ; : : : ; An ) if there exists an injective map : X → {1; 2; : : : ; n} such that x ∈ A(x) for all x ∈ X . The set of partial transversals of A are the independent sets of a matroid on E (see e.g. [8]). The matroid is denoted M (A). If M is an arbitrary matroid and M = M (A), we call M a transversal matroid and A a presentation of M. As shown by Matthews [7] bicircular matroids are transversal matroids. Let Ei be the set of edges of G which are incident with vertex i (i = 1; 2; : : : ; n). Then B(G) is the transversal matroid whose independent sets are the partial transversals of the family of sets E(G) = (E1 ; E2 ; : : : ; En ). The family of sets E(G) is the natural presentation of B(G) corresponding to the graph G. Let A = (A1 ; A2 ; : : : ; An ) be an arbitrary family of subsets of a set E. If M (A) has rank k and (Ai1 ; Ai2 ; : : : ; Aik ) (1 6 i1 ¡ · · · ¡ ik 6 n) has a transversal, then M (A) = M ((Ai1 ; Ai2 ; : : : ; Aik )) [5]. For this reason, we generally assume that (A1 ; A2 ; : : : ; An ) has a transversal. That is, M (A) has rank n. In general, there are many families B = (B1 ; B2 ; : : : ; Bn ) of subsets of E such that M = M (B). It is well-known (shown in [2]) that there exists a unique maximal presentation 2 (M1 ; M2 ; : : : ; Mn ) of M. Here maximal means that for each i and for each x ∈ Mi , (M1 ; : : : ; Mi−1 ; Mi ∪ {x}; Mi+1 ; : : : ; Mn ) is not a presentation of M. The transversal matroid M has, in general, many minimal presentations (C1 ; C2 ; : : : ; Cn ). The sets Ci in each minimal presentation are distinct cocircuits of M. Conversely, a presentation consisting of cocircuits is minimal. Note that if e is a coloop of M, then {e} is a cocircuit and is a set in every minimal presentation of M. Two presentations A = (A1 ; A2 ; : : : ; An ) and B = (B1 ; B2 ; : : : ; Bn ) are isomorphic, written A ∼ = B, if there exists a bijection : {1; 2; : : : ; n} → {1; 2; : : : ; n} such that Ai = B(i) for all i. We say that a presentation A = (A1 ; A2 ; : : : ; An ) is contained in a presentation B = (B1 ; B2 ; : : : ; Bn ) if there exists a bijection : {1; 2; : : : ; n} → {1; 2; : : : ; n} such that Ai ⊆ B(i) for all i ∈ {1; 2; : : : ; n}. 2
Apart from the order of the sets.
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In this paper, we discuss the graphs that give rise to the same bicircular matroid. We give a characterization of bicircular matroids in terms of the presentations. We then show how, given any presentation of a bicircular matroid, to .nd a graph representing the matroid, and that, in some cases, there is more than one such graph. Once we .nd the graph, by the results of Brualdi and Neudauer [4] we can .nd the other minimal presentations, and then other graphs representing this bicircular matroid. We show that a presentation with the property that the intersection of any three sets is empty is the natural presentation for some graph G that represents B(G). By .nding all such presentations, we show that we can .nd all graphs representing the bicircular matroid. In general, we consider the various graphs that can represent one bicircular matroid, and how those graphs are related to each other. In the .nal section, we show that if we use the methods in [4] to .nd the minimal presentations of B(G) that have empty intersection of any three sets, the graphs G that arise from these new presentations are related to G by the operations described in [6].
2. Basic properties Let G be a graph with vertex set V = {1; 2; : : : ; n} and edge set E. Since the bicircular matroid B(G) is the direct sum of the bicircular matroids of its connected components, we assume that G is connected. If G has a pendant edge e, then e is in every basis of B(G) and hence is a coloop of B(G). Henceforth, we assume that every vertex of G is incident with at least two edges, that is, the minimum degree of G is at least 2. We .rst identify the cocircuits of the bicircular matroid B(G) (proved in [4]). Lemma 1. A subset C of the edges E of G is a cocircuit of B(G) if and only if the spanning subgraph G\C obtained from G by deleting the edges in C has exactly one tree component T and each edge in C is incident with at least one vertex of T . The following lemma is proved in [7]. Lemma 2. Let G be a connected graph with at least n + 1 edges. Then B(G) is a connected matroid if and only if G has no pendant edges. In particular; if G has no pendant edges; then each edge of G is in a bicycle. A graph is called bicyclic provided each edge is in a bicycle. Thus a graph is bicyclic if and only if its bicircular matroid has no coloops. Given any graph G, we can easily .nd a presentation of B(G), the natural presentation E(G). The natural presentation has the property that the intersection of any three sets is empty because no edge is adjacent to three vertices of the graph. Each edge is in at most two sets of the natural presentation (one only if the edge is a loop). A presentation with the property that the intersection of any three sets is empty is said to have the triple intersection property. Given an arbitrary presentation of a
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Fig. 2. Wheel on six vertices.
transversal matroid, M, we can determine if M is a bicircular matroid using the following characterization of Matthews [7]. Theorem 3. Bicircular matroids are precisely the loopless transversal matroids which have a presentation such that the intersection of any three members is empty. A natural question that arises is whether every minimal presentation of a bicircular matroid has the triple intersection property. We can see with the example of a wheel centered at vertex 1, given in Fig. 2, that this is not true. Using the results of Brualdi and Neudauer in [4], all the minimal presentations for the graph can be determined. In the case of the wheel in Fig. 2, the natural presentation is a minimal presentation, and the natural presentation has the triple intersection property. We can .nd another minimal presentation of B(W ) by replacing E1 with a set of edges that is the complement of a spanning tree of W [4]. So, another minimal presentation for B(W ) is A(W ) = (C1 ; E2 ; E3 ; E4 ; E5 ; E6 ; E7 ), where C1 = {a; b; f; j; k; l}, E2 = {a; g; h}, E3 = {b; h; i}; E4 = {c; i; j}; E5 = {d; j; k}; E6 = {e; k; l}; E7 = {f; l; g}. This presentation has three sets with non-empty intersection. We see that it is possible to have one minimal presentation with the triple intersection property and another without this property. We know by Theorem 3 that at least one presentation of a bicircular matroid has the triple intersection property. If this presentation is not minimal, it must contain a minimal presentation, and this minimal presentation also has this property. Thus, at least one of the minimal presentations of a bicircular matroid will have the triple intersection property. If M is a bicircular matroid and G is a graph such that M = B(G) then G is a representation of M, and we say that G represents M. If G is a graph with minimum degree at least three and parallel edges allowed there can be more than one graph representing B(G). In the following example, the graphs G in Fig. 3 and G in Fig. 4 represent the same bicircular matroid. The natural presentation E(G) = (E1 ; E2 ; E3 ; E4 ), where E1 = {a; b; c}; E2 = {a; b; d}; E3 = {c; d; e; f; g}, E4 = {e; f; g} is not a minimal presentation for the matroid.
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Fig. 3. G represents B(G).
Fig. 4. G also represents B(G).
As shown in [4], we can .nd a minimal presentation for B(G) by removing from E3 one edge from vertex 3 to the tree component of G3 , which is vertex 4. This gives us a minimal presentation with the triple intersection property, which is the natural presentation with respect to G . Here E(G ) = (E1 ; E2 ; E3 ; E4 ), where E1 = {a; b; c}, E2 = {a; b; d}, E3 = {c; d; e; g}, E4 = {e; f; g}. Lemma 4. Let G be a graph; and let E(G) be the natural presentation of the bicircular matroid of G. Then G is determined up to isomorphism by E(G). Proof. Let G and H be two arbitrary graphs with vertices {1; 2; : : : ; n} such that G = H . Since they are not the same graph, there is some edge e incident with vertex v in G that is not incident with vertex v in H . In a natural presentation, ej ∈ Ej if and only if ej is incident with vertex j. So e ∈ Ev in the natural presentation of B(G), but e ∈ Ev in the natural presentation of B(H ), and the natural presentations are not the same. So the map from the natural presentation to the graph is well de.ned. Let G be a graph with natural presentation E(G) = (E1 ; E2 ; : : : ; En ), and H be a graph with natural presentation E(H ) = (Ei1 ; Ei2 ; : : : ; Ein ). Assume G ∼ = H . Then there are bijections : V (G) → V (H ) and % : E(G) → E(H ) such that the edge e of G is incident with a vertex v of G if and only if %(e) is incident with (v). Then : j → ij and e ∈ Ej if and only if %(e) ∈ E ( j) , and E(G) ∼ = E(H ). Assume E(G) ∼ = E(H ). Then there exist bijections & : E(G) → E(H ) and : j → ij such that &(Ej ) = E( j) and e ∈ Ej if and only if &(e) ∈ E( j) . Then e is incident with vertex j of G if and only if &(e) is incident with (j), and G ∼ = H.
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Hereafter if we say G = H for two graphs G and H , we mean up to relabelling of the vertices and the edges. The following theorem follows from [4] and from Lemma 4. Theorem 5. Let G be a graph with no parallel edges and minimum degree at least three. If G is a graph such that B(G) = B(G ); then G ∼ = G . Proof. If G is such a graph, the natural presentation is a minimal presentation. The only set in any presentation that diLers from a set of the natural presentation is Ei , where i is the center of a wheel [4]. In this case, we obtain another minimal presentation by replacing the edges of Ei that are spokes of the wheel with the complement of a spanning tree of the wheel. Then any such cocircuit Ci that diLers from Ei has in it some edge from the rim of the wheel. Each rim edge is necessarily in at least two sets of any presentation. For, if {j; k} is a rim edge, it is in Ej and Ek . G\Ej has no coloops in the matroid B(G\Ej ) and G\Ek has no coloops in the matroid B(G\Ek ), so Ej and Ek are sets in every presentation of B(G). If {j; k} is also in Ci , it is in three sets of this minimal presentation. Thus no other presentation has the triple intersection property. If there were another graph, G , representing B(G), there would be a presentation of B(G) with the triple intersection property which is the natural presentation of B(G) with respect to G . By Lemma 4, the natural presentation determines the graph up to isomorphism. So, G ∼ = G . So, if G has no parallel edges and minimal degree at least three, there is a unique graph, up to isomorphism, representing B(G). If G is a graph with minimum degree at least three and parallel edges allowed, there can be more than one graph representing B(G). Coullard et al. [6] characterize those graphs that represent the same bicircular matroid, and we discuss their work in Section 4.
3. Construction of the graphs from a presentation Given an arbitrary presentation of a transversal matroid, M, we can determine if M is a bicircular matroid: from an arbitrary presentation we can .nd the maximal presentation by an algorithm of Brualdi and Dinolt [3], and from this we can .nd the other presentations of the matroid by Bondy and Welsh [1]. If there is a presentation with the triple intersection property, we can .nd it using these results. This allows us to determine, although not eMciently, whether a transversal matroid is a bicircular matroid. If M is a bicircular matroid, we can .nd a graph representing M. In general, there is not a unique graph representing the matroid. Algorithm 6. Given any presentation for a bicircular matroid M the graphs G representing M such that B(G) = M can be determined as follows: (1) Determine all the presentations of M (see [1]).
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(2) For each presentation A with the intersection of any three sets empty; construct a graph G from A by putting |Ai ∩ Aj | edges between vertex i and vertex j. If an edge appears in only one set; put a loop at that vertex. Notice that the presentation, A, with the triple intersection property is the natural presentation for B(G) with respect to G. Theorem 7. Given any presentation for a bicircular matroid M; then; up to isomorphism; Algorithm 6 determines all graphs G representing M such that B(G) = M. Proof. Suppose G represents M and E(G) is a natural presentation of M. By Lemma 4, a natural presentation determines the graph, up to isomorphism. A natural presentation always has the triple intersection property. So the algorithm produces all graphs representing M. Coullard et al. [6] showed Lemma 8. Let G be a graph such that B(G) is connected. Then there exists a graph representation H of B(G) such that the star of every vertex of H is a cocircuit of B(G). Here the star of a vertex is the set of edges incident with that vertex. Corollary 9. Let M be a bicircular matroid and let A be a minimal presentation of M with the triple intersection property. Then there exists a graph representation H of M such that every cocircuit of A is the star of a vertex of H . Proof. A is a minimal presentation of M such that the intersection of any three sets of A is empty. Using Algorithm 6 we can construct a graph representing M with A as its natural presentation. Each cocircuit, Ai , of A is composed of the edges of the graph incident with vertex i. So the set of edges Ai is the star of vertex i. From a presentation of a bicircular matroid we can thus .nd the collection of graphs representing a bicircular matroid. In the next section we describe how Coullard et al. [6] characterized the collection of graphs having the same bicircular matroid as that of a given graph. 4. Graphs that represent the same bicircular matroid A natural question that arises from the preceding discussion is how graphs G and G are related if B(G) is also represented by G . In this section we will discuss some results of Coullard et al. [6] in which they characterize the collection of graphs that represent a given bicircular matroid.
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Fig. 5. Balloons of G.
We begin with some notation and de.nitions. In [9] Wagner discusses matroid connectivity further. A cut vertex of a connected graph is one whose deletion produces a disconnected graph. Let F be a proper subset of E, the edges of the graph G, and let H be the subgraph of G induced by F. The set of vertices common to H and to G[E −E(H )] are the vertices of attachment of H . A block of G is a maximal subgraph satisfying the property that every pair of edges is contained in a cycle. An end-block of G is a block H having exactly one vertex of attachment, called the tip of H . Coullard et al. [6] de.ne a balloon of G as a maximal set B of edges such that the subgraph induced by B is a subdivision of one of the graphs in Fig. 5, and the only vertex of attachment of the balloon is v as indicated in the .gure. A line is a set of edges not contained in any balloon, that forms a path, the internal vertices of which have degree two in G and the end vertices of which have degree at least three. Following is a description of operations from [6] which, when applied to a graph, produce a graph with the same bicircular matroid. Let S be a line of G having end vertices u and v, and let e be the unique edge of S incident to v. Rede.ne the incidence relation of e so that e is incident to a vertex w = v of S instead of v. This new graph, G is obtained from G by a rolling of S away from v, and G is obtained from G by an unrolling of S to v. S is a balloon of G . The following two theorems were .rst proved by Wagner [9] for the case where B(G) is 3-connected, and later by Coullard et al. [6] when B(G) is connected. Theorem 10. Let G and G be graphs such that B(G) is connected and G is obtained from G by a rolling of a line S away from one of its end vertices v. Then B(G) = B(G ) if and only if there exists an end-block H of G such that S ⊆ E(H ); v is a tip of H; and every cycle of H contains v. Let v be a vertex incident to exactly three lines L1 , L2 , and L3 in G. Suppose the other end vertex of L1 is u, and the other end of L2 and L3 is w = u. Let e1 be the edge of L1 incident to u, and let e2 be the edge of L2 incident to w. Rede.ne the incidence relations of e1 and e2 so that e1 is incident to w instead of u, and e2 is incident to u instead of w. This new graph, G is obtained from G by a rotation of L1 and L2 at the vertex v. Theorem 11. With the above notation; let G and G be graphs such that B(G) is connected and G is obtained from G by a rotation of lines L1 and L2 at v; an end
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vertex incident with both L1 and L2 . Then B(G) = B(G ) if and only if there exists 3 an endblock H of G such that L:= i=1 Li is contained in E(H ); u is a tip of H and every cycle of H \L contains u; where L3 is the line having the same ends as L2 and u is the end of L1 not equal to v. Let S be a line (respectively, balloon) of G. Let G be a graph obtained from G by replacing S with another line (respectively, balloon) on the same edge set and having the same vertices (respectively, vertex) of attachment. Then G is obtained from G by replacement. Also, if G and G are graphs on the same edge set and each graph is a bicycle, then G is obtained from G by replacement. Theorem 12. Let G and G be graphs such that G is obtained from G by replacement. Then B(G) = B(G ). Rollings, unrollings, rotations, and replacements are operations. If G is obtained from G by an operation and B(G) = B(G ), we say that the operation is legitimate, and that G is r-equivalent to G. We de.ne two graphs to be b-equivalent if they have the same bicircular matroid. Thus r-equivalence implies b-equivalence. The converse is true if we rule out some small cases. Coullard et al. [6] proved. Theorem 13. Let G and G be b-equivalent graphs such that B(G) is connected and |V (G)| = |V (G )| ¿ 5. Then G and G are r-equivalent.
5. Graphs that arise from the minimal presentations of a bicircular matroid Brualdi and Neudauer [4] characterized the minimal presentations of a bicircular matroid B(G) in terms of the graph G. Earlier in this paper we showed that a presentation with the triple intersection property is the natural presentation for some graph G that represents B(G). Here we show that if we use the methods in [4] to .nd the minimal presentations of B(G) with the triple intersection property, the graphs G that have these new presentations as natural presentations are related to G by the operations developed by Coullard et al. [6] and described in Section 4. We want to show that every minimal presentation with the triple intersection property is the natural presentation for a graph that is related to our original graph by a series of legitimate operations described in Section 4. Throughout this section, G is a connected graph having minimum degree at least two and having at least .ve vertices. Lemma 14. Let G be a graph such that the natural presentation of B(G) is not minimal. Then there is a minimal presentation of B(G) contained in the natural presentation E(G). The graph that arises from this minimal presentation is obtained from G by either a series of replacements of balloons or rolling away of lines; and is r-equivalent to G.
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Proof. The natural presentation is not minimal if and only if there is a set Ei in the natural presentation whose removal leaves at least two trees by Lemma 1. Suppose G \ Ei has exactly two acyclic components for some i. One acyclic component is i, and i is the tip of an endblock containing the other tree. Every cycle in this endblock passes through i. Since Ei is not a cocircuit, by the results of [4], there is an edge {i; j} such that Ei \ {i; j} is a cocircuit. This edge was either in a balloon of G with vertex of attachment i, or it was in a line of G that has as one end vertex i. We have a new presentation with Ei replaced by Ci = Ei \{i; j}, and this presentation has the triple intersection property. The edge we removed from Ei now appears in only one set of the presentation, Ej . We construct a graph from this presentation by putting |Ck ∩ Cl | edges between vertex k and l, for each pair of vertices. There is a loop at vertex j. This new graph, G , is obtained from G either by the replacement of a balloon, if {i; j} was in a balloon in G, or by the rolling away of a line from vertex i, if {i; j} was in a line in G. Thus, by Theorem 10, G is obtained from G by a legitimate operation, and G is r-equivalent to G. We repeat this for each acyclic component of G \ Ei other than {i} and for each vertex i for which Ei is not a cocircuit. In this way we construct a graph H that is obtained from G by a series of replacements of balloons and rolling away of lines, and is r-equivalent to G. We have B(G) = B(H ), and the natural presentation with respect to H is a minimal presentation for B(G). We now have a graph H that represents B(G) whose natural presentation, E(H ), is a minimal presentation. We now show that there may be a minimal presentation for B(G) that is diLerent from the natural presentation when G\Ei has a cycle component with (possibly trivial) trees attached. We de.ne a wheel with possibly multiple edge spokes to be a pinwheel. Notice that a wheel is a pinwheel, but not conversely. Lemma 15. Let G be a graph with minimum degree at least three; and let i be a vertex of G such that G\Ei has a component that is a cycle. Then there is a minimal presentation of B(G) with the triple intersection property such that the graph; G ; that arises from this presentation is obtained from G by rotation or by a series of unrolling and rolling away of lines; and G is r-equivalent to G. Proof. If G is a graph with minimum degree at least three and no parallel edges, then since there is a component of G\Ei that is a cycle, there is a wheel centered at vertex i. By Theorem 5, we know there is a unique 3 graph representing B(G), and none of the other presentations has the triple intersection property. Suppose G has minimum degree at least three with parallel edges. If G \ Ei has a cycle component, then either i is the center of a pinwheel or the cycle component consists of (i) two edges on two vertices or (ii) a single loop at some vertex j. 3
Up to isomorphism.
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If i is the center of a pinwheel, as in case of a wheel, which is shown in the proof of Theorem 5, the only presentation of this portion of the graph with the triple intersection property is the natural presentation. In any other presentation, we must use an edge from the rim of the pinwheel, and this edge is necessarily in two other sets of every presentation. If the cycle is two edges on two vertices, i is the tip of an endblock F. Notice 3 that this endblock, F, consists of several lines. L:= i=1 Li is contained in E(F), v is the vertex incident with all three lines, i is a tip of F and every cycle of F \ L contains i, where L3 is the line having the same ends as L2 and i is the end of L1 not equal to v. The other minimal presentations are composed of distinct cocircuits of the matroid. If we remove an edge e from Ei , we must replace it with an edge from the cycle [4]. We must also remove the edge we have put into Ei from one of the sets so that it is not in three sets of the presentation, and we must replace this with e. We can do this only if edge e is a coloop of the matroid B(G \ Ej ), where j is the other end vertex of lines L2 and L3 . Now we have a presentation that is made up of cocircuits with the triple intersection property. We have exchanged two edges between two sets of the presentation. We construct the graph from this new minimal presentation, and see that it is obtained from G by a rotation of L1 with one of the other lines at v. By Theorem 11 this is a legitimate operation, and the new graph is r-equivalent to G. If the cycle is the single loop at j, we can replace an edge {i; j} in Ei with the loop edge. Now edge {i; j} appears in only one set of the presentation, Ej , and the loop edge appears in Ei and Ej . When we construct the new graph from this presentation, these two edges have switched roles. This new graph is obtained from G by the unrolling of one line (the loop) and then the rolling away of another line (edge {i; j}). Since i is the tip of an endblock and every cycle of that endblock contains i, by Theorem 10 this is a legitimate operation, and the new graph is r-equivalent to G. Note that in this case we can add the loop edge to Ei without removing the edge {i; j}. In this case, the new graph is obtained by the unrolling of a line (the loop), and is also r-equivalent to G. Lemma 16. Let G be a graph for which the natural presentation of B(G) is a minimal presentation; and let i be a vertex of degree two. There is another minimal presentation of B(G) with the triple intersection property that replaces the set of the natural presentation Ei with another cocircuit. The graph that arises from this presentation is obtained from G by the replacement of a line by a line or the replacement of a balloon by a balloon; and is r-equivalent to G. Proof. Because the natural presentation of B(G) is a minimal presentation, all degree two vertices are either contained in a line or in the part of a balloon that is not a cycle. The cycle portion of a balloon is a single edge loop. The other minimal presentations of B(G) are obtained by letting the edges of the line or balloon be the vertices of a tree, and each pair of connected vertices is a cocircuit
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of the minimal presentation [4]. Since the presentation must have the triple intersection property, this tree has maximum degree two, and so is a path. Let u be a vertex of degree at least three that is adjacent to a degree two vertex and is not at the loop of a balloon. Every edge of the line or balloon is a coloop of G \ Eu . We can replace the edge of the line or balloon that is in Eu with any other edge of the line or balloon. To ensure that the presentation has the triple intersection property, we put in Eu an edge that is one of the pendant vertices of the tree above. We construct the graph from this new presentation. Every edge that was in a line is still in a line, and every edge that was in a balloon is still in a balloon, with the same vertices or vertex of attachment. The new graph is obtained from G by the replacement of a line by a line, or of a balloon by a balloon. By Theorem 12 this is a legitimate operation, and thus the new graph is r-equivalent to G. These three lemmas encompass the cases where the removal of a set of edges about a vertex yields some coloops, and thus there are other presentations of the matroid. This is all tied together in the following theorem. This theorem follows from Theorem 13. In fact, we could state and prove a stronger result here by removing the word “minimal”. We instead prove the following result because, as such, the proof of Theorem 17 and the preceding lemmas demonstrate the correlation between the techniques of Brualdi and Neudauer [4] and those of Coullard et al. [6]. Theorem 17. Let G be a connected graph having minimum degree at least two and having at least ve vertices. Every minimal presentation for B(G) with the triple intersection property is the natural presentation for a graph that is related to G by a series of legitimate operations; and this new graph is r-equivalent to G. Proof. If G is a graph with no parallel edges and minimum degree at least three, then up to isomorphism there is only one graph representing B(G) by Theorem 5. There is only one presentation of B(G) with the triple intersection property. If G is a graph with minimum degree at least three, for each vertex i, each of the components of Gi is one of the following three types: (x) a tree, (y) a cycle with a (possibly trivial) tree rooted at each vertex, and (z) a bicyclic graph with a (possibly trivial) tree rooted at each vertex. Case 1: i is a vertex of type (x). In this case the natural presentation is not minimal. Lemma 14 shows that we can .nd the minimal presentations with the triple intersection property. The graphs that arise from these presentations are obtained from G by either a series of replacements of balloons, or the rolling away of a lines, and are r-equivalent to G. Case 2: i is a vertex of type (y). While the tree rooted at each vertex does provide coloops of the matroid B(G \ Ei ), these edges do not replace any edges in Ei . The edges of these trees are already in two sets of the presentation. If j is a vertex of one
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of these trees, G \ Ej has no coloops in the matroid B(G \ Ej ). So the edges of those trees are always in the same two sets of the presentation, and cannot be added to Ei . By Lemma 15 we know that if G \ Ei has a component that is a cycle, there is another minimal presentation of B(G) with the triple intersection property, and the graph, G , that arises from this presentation is obtained from G by rotation or by a series of unrolling and rolling away of lines, and G is r-equivalent to G. Since the trees rooted at the vertices of the cycle do not provide any other minimal presentation with the triple intersection property, these are all the graphs we get in this case. Case 3: i is a vertex of type (z). We showed above that the trees rooted at each vertex do not provide additional minimal presentations with the triple intersection property. There are no other coloops of B(G \ Ei ). So, there are no graphs obtained from G by replacing Ei with another cocircuit, where i is a vertex of this type. If G is a graph with minimum degree at least two, there is a unique graph G ∗ , called the mothergraph of G, such that G ∗ has minimum degree at least three and G is a subdivision of G ∗ . Brualdi and Neudauer [4] characterized the minimal presentations of B(G) in terms of the minimal presentations of B(G ∗ ). By the above results we see how the graph constructed from the other minimal presentations of B(G ∗ ) is related to G ∗ , and then to G. We still need to consider the vertices of degree two. If the natural presentation for B(G) is not a minimal presentation, we can .nd a minimal presentation by Lemma 14. The graph, G , that arises from this presentation is r-equivalent to G. We can apply Lemma 16 to G , which also represents B(G). This lemma tells us that if i is a vertex of degree two and we .nd another minimal presentation of B(G) with the triple intersection property, the graph that arises from this presentation, G is obtained from G by the replacement of a line by a line, or the replacement of a balloon by a balloon, and is r-equivalent to G . Since G is r-equivalent to G, and G is r-equivalent to G , G is r-equivalent to G. Acknowledgements I am grateful to the referees for their detailed suggestions. References [1] J.A. Bondy, D.J.A. Welsh, Some results on transversal matroids and constructions for identically self-dual matroids, Quart. J. Math. Oxford (2) 22 (1971) 435–451. [2] R.A. Brualdi, Transversal matroids, in: N. White (Ed.), Combinatorial Geometries, Cambridge Univ. Press, Cambridge, 1985. [3] R.A. Brualdi, G.W. Dinolt, Characterizations of transversal matroids and their presentations, J. Combin. Theory 12 (1972) 268–286. [4] R.A. Brualdi, N.A. Neudauer, The minimal presentations of a bicircular matroid, Quart. J. Math. Oxford (2) 48 (1997) 17–26. [5] R.A. Brualdi, E.B. Scrimger, Exchange systems, matchings, and transversals, J. Combin. Theory 5 (1968) 244–257.
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[6] C.R. Coullard, J.G. del Greco, D.K. Wagner, Representations of bicircular matroids, Discrete Appl. Math. 32 (1991) 223–240. [7] L.R. Matthews, Bicircular matroids, Quart. J. Math. Oxford (2) 28 (1977) 213–228. [8] J. Oxley, Matroid Theory, Oxford University Press, Oxford, 1992. [9] D.K. Wagner, Connectivity in bicircular matroids, J. Combin. Theory Ser. B 39 (1985) 308–324.