How to build a brick Original Research Article

A 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 image of G, the graph image 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 image 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 image. 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 image, image 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) image is a matching covered graph with image and (ii) for each barrier B of image, the graph image has precisely image isolated vertices, each of which has degree two in image. 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.