Extremal critically connected matroids

AbstractA connected matroid M is called a critically connected matroid if the deletion of any one element from M results in a disconnected matroid. We show that a critically connected matroid of rank n, n≥3, can have at most 2n−2 elements. We also show that a critically connected matroid of rank n on 2n−2 elements is isomorphic to the forest matroid of K2, n−2

How to build a brick

AbstractA graph is matching covered if it connected, has at least two vertices and each of its edges is contained in a perfect matching. A 3-connected graph G is a brick if, for any two vertices u and v of G, the graph G-{u,v} has a perfect matching. As shown by Lovász [Matching structure and the matching lattice, J. Combin. Theory Ser. B 43 (1987) 187–222] every matching covered graph G may be decomposed, in an essentially unique manner, into bricks and bipartite graphs known as braces. The number of bricks resulting from this decomposition is denoted by b(G).The object of this paper is to present a recursive procedure for generating bricks. We define four simple operations that can be used to construct new bricks from given bricks. We show that all bricks may be generated from three basic bricks K4, C¯6 and the Petersen graph by means of these four operations. In order to establish this, it turns out to be necessary to show that every brick G distinct from the three basic bricks has a thin edge, that is, an edge e such that (i) G-e is a matching covered graph with b(G-e)=1 and (ii) for each barrier B of G-e, the graph G-e-B has precisely |B|-1 isolated vertices, each of which has degree two in G-e. Improving upon a theorem proved in [M.H. de Carvalho, C.L. Lucchesi, U.S.R. Murty, On a conjecture of Lovász concerning bricks, I, The characteristic of a matching covered graph, J. Combin. Theory Ser. B 85 (2002) 94–136; M.H. de Carvalho, C.L. Lucchesi, U.S.R. Murty, On a conjecture of Lovász concerning bricks, II, Bricks of finite characteristic, J. Combin. Theory Ser. B 85 (2002) 137–180] we show here that every brick different from the three basic bricks has an edge that is thin.A cut of a matching covered graph G is separating if each of the two graphs obtained from G by shrinking the shores of the cut to single vertices is also matching covered. A brick is solid if it does not have any nontrivial separating cuts. Solid bricks have many interesting properties, but the complexity status of deciding whether a given brick is solid is not known. Here, by using our theorem on the existence of thin edges, we show that every simple planar solid brick is an odd wheel

Optimal Ear Decompositions Of Matching Covered Graphs And Bases For The Matching Lattice

This is a sequel to our papers (M. H. de Carvalho, C. L. Lucchesi, and U. S. R. Murty, 1999, Combinatorica 19, 151-174; 2002, J. Combin. Theory Ser. B 85, 94-136; and 2002, J. Combin. Theory Ser. B 85, 137-180). A Petersen brick is a graph whose underlying simple graph is isomorphic to the Petersen graph. For a matching covered graph G, b(G) denotes the number of bricks of G, and p(G) denotes the number of Petersen bricks of G. An ear decomposition of G is optimal if, among all ear decompositions of G, it uses the least possible number of double ears. Here we make use of the main theorem in (2002, J. Combin. Theory Ser. B 85, 137-180) to prove that the number of double ears in an optimal ear decomposition of a matching covered graph G is b(G) + p(G). In particular, if G is a brick that is not a Petersen brick, then there is an ear decomposition of G with exactly one double ear. This answers a question raised by D. Naddef and W. R. Pulleyblank (1982, Ann. Discrete Math. 16, 241-260). Using this theorem, we give an alternative proof of L. Lovász' matching lattice characterization theorem (1987, J. Combin. Theory Ser. B 43, 187-222). We also show that for any matching covered graph G, there is a basis for the matching lattice of G consisting of incidence vectors of perfect matchings of G. This answers a question raised by U. S. R. Murty (1994, "The Matching Lattice and Related Topics," Technical Report, University of Waterloo). In fact, we show that such a basis may be obtained from the incidence vectors of perfect matchings associated with optimal ear decompositions of G. © 2002 Elsevier Science (USA).8515993de Carvalho, M.H., "Decomposição Ótima em Orelhas para Grafos Matching Covered" (1997), Ph.D. thesis, Institute of Computing, University of Campinas, Brazil, [In Portuguese]de Carvalho, M.H., Lucchesi, C.L., Murty, U.S.R., Ear decompositions of matching covered graphs (1999) Combinatorica, 19, pp. 151-174de Carvalho, M.H., Lucchesi, C.L., Murty, U.S.R., On a conjecture of Lovász concerning bricks, I. The characteristic of a matching covered graph (2002) J. Combin. Theory Ser. B, 85, pp. 94-136de Carvalho, M.H., Lucchesi, C.L., Murty, U.S.R., On a conjecture of Lovász concerning bricks, II. Bricks of finite characteristic (2002) J. Combin. Theory Ser. B, 85, pp. 137-180Edmonds, J., Lovász, L., Pulleyblank, W.R., Brick decomposition and the matching rank of graphs (1982) Combinatorica, 2, pp. 247-274Lovász, L., Matching structure and the matching lattice (1987) J. Combin. Theory Ser. B, 43, pp. 187-222Lovász, L., Plummer, M.D., "Matching Theory" (1986) Annals of Discrete Mathematics, 29. , Elsevier Science, AmsterdamMurty, U.S.R., The Matching Lattice and Related Topics (1994), Technical report, University of Waterloo, Preliminary ReportNaddef, D., Rank of maximum matchings in a graph (1982) Math. Programming, 22, pp. 52-70Naddef, D., Pulleyblank, W.R., Ear decomposition of elementary graphs and GF(2)-rank of perfect matchings (1982) Ann. Discrete Math, 16, pp. 241-260. , Bonn Workshop on Combinatorial Optimization, A. Bachem (M. Grötschel and B. Korte, Eds.)Schrijver, A., "Theory of Linear and Integer Programming" (1986), WileySeymour, P.D., On multicolourings of cubic graphs and conjectures of Fulkerson and Tutte (1979) Proc. London Math. Soc. Ser. 3, 38, pp. 423-460Szigeti, Z., The two ear theorem on matching-covered graphs (1998) J. Combin. Theory Ser. B, 74, pp. 104-109Szigeti, Z., Perfect matchings versus odd cuts submitted for publicatio

On The Number Of Dissimilar Pfaffian Orientations Of Graphs

The dissimilar Pfaffian orientations in planar and biparite graphs are discussed. An orientation D of graph G is Pfaffian if the number of edges of circuit C whose directions in D agree with any prescribed sense of orientation of c is odd. The properties of the matching covered graphs are used to study Pfaffian orientations of the graphs. A property of minimal graphs without a Pfaffian orientation is established and use it to give an alternative proof of the characterization of Pfaffian biparite graphs.39193113De Carvalho, M.H., Lucchesi, C.L., Murty, U.S.R., The perfect matching polytope and solid bricks (2004) J. Combin. Theory B, 92, pp. 319-324De Carvalho, M.H., Lucchesi, C.L., Murty, U.S.R., Ear decompositions of matching covered graphs (1999) Combinatorial, 19, pp. 151-174De Carvalho, M.H., Lucchesi, C.L., Murty, U.S.R., On a conjecture of Lovász concerning bricks. I. The characteristic of a matching covered graph (2002) J. Comb. Theory B, 85, pp. 94-136De Carvalho, M.H., Lucchesi, C.L., Murty, U.S.R., On a conjecture of Lovász concerning bricks. II. Bricks of finite characteristic (2002) J. Comb. Theory B, 85, pp. 137-180De Carvalho, M.H., Lucchesi, C.L., Murty, U.S.R., Optimal ear decompositions of matching covered graphs (2002) J. Comb. Theory B, 85, pp. 59-93Edmonds, J., Lovász, L., Pulleyblank, W.R., Brick decomposition and the matching rank of graphs (1982) Combinatorica, 2, pp. 247-274Fischer, I., Little, C.H.C., A characterisation of Pfaffian near bipartite graphs (2001) J. Comb. Theory B, 82, pp. 175-222Kasteleyn, P.W., Dimer statistics and phase transitions (1963) J. Math. Phys., 4, pp. 287-293Little, C., A characterization of convertible (0, 1)-matrices (1975) J. Comb. Theory B, 18, pp. 187-208Little, C.H.C., Rendl, F., Operations preserving the Pfaffian property of a graph (1991) J. Austral. Math. Soc. Ser. A, 50, pp. 248-275Lovász, L., Matching structure and the matching lattice (1987) J. Comb. Theory B, 43, pp. 187-222Lovász, L., Plummer, M.D., (1986) Matching Theory, 29. , Annals of Discrete Mathematics. Elsevier ScienceMcCuaig, W., Brace generation (2001) J. Graph Theory, 38, pp. 124-169Robertson, N., Seymour, P.D., Thomas, R., Permanents, Pfaffian orientations and even directed circuits (1999) Ann. Math., 150, pp. 929-975Tutte, W.T., Graph theory as I have known it (1998) Oxford Lecture Ser. Math. Appl., 11. , Clarendon Press, OxfordVazirani, V.V., Yanakakis, M., Pfaffian orientation of graphs, 0,1 permanents, and even cycles in digraphs (1989) Discrete Appl. Math., 25, pp. 179-18

On A Conjecture Of Lovász Concerning Bricks. I. The Characteristic Of A Matching Covered Graph

In 1987, Lovász conjectured that every brick G different from K4, C̄6, and the Petersen graph has an edge e such that G-e is a matching covered graph with exactly one brick. Lovász and Vempala announced a proof of this conjecture in 1994. Their paper is under preparation. In this paper and its sequel (M. H. de Carvalho, C. L. Lucchesi, and U. S. R. Murty, 2002, J. Combin. Theory Ser. B 85, 137-180) we present a proof of this conjecture. We shall in fact prove that if G is a brick different from K4,C̄6, R8 that does not have the Petersen graph as its underlying simple graph, then it has two edges e and f such that both G-e and G-f are matching covered graphs with exactly one brick, with the additional property that, in each case, the underlying simple graph of that one brick is different from the Petersen graph. A cut C of a matching covered graph G is a separating cut if the two C-contractions of G are matching covered. In this paper, we introduce the notion of the characteristic of a separating cut in a matching covered graph and establish some basic properties. We use those properties to first prove our theorem for solid bricks, that is, bricks which do not have any nontrivial separating cuts. The proof of the theorem for nonsolid bricks will be presented in the sequel. © 2002 Elsevier Science (USA).85194136de Carvalho, M.H., "Decomposição Ótima em Orelhas para Grafos Matching Covered" (1996), Ph.D. thesis, submitted to the University of Campinas, Brazil, [In Portuguese]de Carvalho, M.H., Lucchesi, C.L., Edge implication in matching covered graphs (1996), pp. 47-55. , "Anais do Workshop Internacional de Combinatória," Federal University of Rio de Janeirode Carvalho, M.H., Lucchesi, C.L., Murty, U.S.R., Ear decompositions of matching covered graphs (1999) Combinatorica, 19, pp. 151-174de Carvalho, M.H., Lucchesi, C.L., Murty, U.S.R., On a conjecture of Lovász concerning bricks, II. Bricks of finite characteristic (2002) J. Combin. Theory Ser. B, 85, pp. 137-180de Carvalho, M.H., Lucchesi, C.L., Murty, U.S.R., Optimal ear decompositions of matching covered graphs and bases for the matching lattice (2002) J. Combin. Theory Ser. B, 85, pp. 59-93de Carvalho, M.H., Lucchesi, C.L., A family of cubic matching covered graphs (1999), in preparationGerards, A.M.H., Lovász, L., Truemper, K., Schrijver, A., Seymour, P., Shih, S., Regular matroids from graphs in preparationLovász, L., Plummer, M.D., "Matching Theory" (1986), North-Holland, AmsterdamLovász, L., Matching structure and the matching lattice (1987) J. Combin. Theory Ser. B, 43, pp. 187-222Lovász, L., Vempala, S., (1994), unpublished abstractLovász, L., Vempala, S., Removable edges in matching-covered graphs (1998), manuscriptMurty, U.S.R., "The Matching Lattice and Related Topics" (1994), Preliminary Report, University of Waterloo, Waterloo, CanadaNaddef, D., Pulleyblank, W.R., Ear decomposition of elementary graphs and GF(2)-rank of perfect matchings (1982) Ann. Discrete Math, 16, pp. 241-260. , "Bonn Workshop on Combinatorial Optimization" (A. Bachem, M. Grötschel, and B. Korte, Eds.)Szigeti, Z., The two ear theorem on matching-covered graphs (1998) J. Combin. Theory Ser. B, 74, pp. 104-109Szigeti, Z., Perfect matchings versus odd cuts (1998), submitted for publicationTutte, W.T., "Connectivity in Graphs" (1966), University of Toronto Press, Toront

On A Conjecture Of Lovász Concerning Bricks. Ii. Bricks Of Finite Characteristic

In (M. H. de Carvalho, C. L. Lucchesi, and U. S. R. Murty, 2002, J. Combin. Theory Ser. B 85, 94-136) we established the validity of the main theorem (1.1) for solid bricks. Here, we establish the existence of suitable separating cuts in nonsolid bricks and prove the theorem by applying induction to cut-contractions with respect to such cuts. © 2002 Elsevier Science (USA).851137180de Carvalho, M.H., Lucchesi, C.L., Murty, U.S.R., On a conjecture of Lovász concerning bricks, I. The characteristic of a matching covered graph (2002) J. Combin. Theory Ser. B, 85, pp. 94-136de Carvalho, M.H., Lucchesi, C.L., Murty, U.S.R., Ear decompositions of matching covered graphs (1999) Combinatorica, 19, pp. 151-174Lovász, L., Plummer, M.D., "Matching Theory" (1986), North-Holland, AmsterdamCampos, C.N., Lucchesi, C.L., "On the Relation between the Petersen Graph and the Characteristic of Separating Cuts in Matching Covered Graphs" (2000), Technical Report 2000-22, Institute of Computing, University of Campinas, Brazi