Carvalho, Lucchesi and Murty (2006, How to build a brick, Discrete Math., 306, 2383‐2410) gave a generation procedure for bricks. In particular, they showed that every brick may be constructed from K 4, the triangular prism C true¯ 6, and the Petersen graph. The object of this paper is to establish a generation procedure that is specific to the class of near‐bipartite bricks. In particular, we show that every near‐bipartite brick may be constructed from K 4 and C true¯ 6 so that each intermediate brick is also near‐bipartite. Norine and Thomas (2007, Generating bricks, J. Combin. Theory Ser. B, 97, 769‐817) proved a generation theorem for simple bricks. In a subsequent work with Marcelo H. de Carvalho (2017, Generating simple near‐bipartite bricks, https://arxiv.org/abs/1704.08796), we use the results of this paper to prove a generation theorem for simple near‐bipartite bricks.
A bi‐subdivision of a graph J is a graph H obtained from J by subdividing each of its edges by inserting an even number of vertices. A matching covered subgraph H of a matching covered graph G is conformal if G−V(H) has a perfect matching. Using the theory of ear decompositions, Lovász (Combinatorica, 3 (1983), 105–117) showed that every nonbipartite matching covered graph has a conformal subgraph which is either a bi‐subdivision of K4 or of C6¯. (The graph C6¯ is the triangular prism.) A matching covered graph is K4‐based if it contains a bi‐subdivision of K4 as a conformal subgraph; otherwise it is K4‐free. C6¯‐based and C6¯‐free graphs are analogously defined. The result of Lovász quoted above implies that any nonbipartite matching covered graph is either K4‐based or C6¯‐based (or both). The problem of deciding which matching covered graphs are K4‐based and which are C6¯‐based is, in general, unsolved. In this paper, we present a solution to this classification problem in the special case of planar graphs. In Section 2, we show that a matching covered graph is K4‐free (C6¯‐free) if and only if each of its bricks is K4‐free (C6¯‐free). In Section 5, we show that a planar brick is K4‐free if and only if it has precisely two odd faces. In Section 6, we determine the list of all C6¯‐free planar bricks; apart from one exception, it consists of two infinite families of bricks. The principal tool we use for proving our results is the brick generation procedure established by Norine and Thomas (J Combin Theory Ser B, 97 (2007), 769–817).
Lovász (1987) proved that every matching covered graph G may be uniquely decomposed into a list of bricks (nonbipartite) and braces (bipartite); we let b(G) denote the number of bricks. An edge e is removable if G − e is also matchingA brick G is near-bipartite if it has a pair of edges {e, f } so that G − e − f is matching covered and bipartite; such a pair {e, f } is a removable doubleton. (Each of K 4 and the triangular prism C 6 has three removable doubletons.) Carvalho, Lucchesi and Murty (2002) proved a conjecture of Lovász which states that every brick, distinct from K 4 , C 6 and the Petersen graph, has a b-invariant edge.A cubic graph is essentially 4-edge-connected if it is 2-edge-connected and if its only 3-cuts are the trivial ones; it is well-known that each such graph is either a brick or a brace; we provide a graph-theoretical proof of this fact.We prove that if G is any essentially 4-edge-connected cubic brick then its edgeset may be partitioned into three (possibly empty) sets: (i) edges that participate in a removable doubleton, (ii) b-invariant edges, and (iii) quasi-b-invariant edges; our Main Theorem states that if G has two adjacent quasi-b-invariant edges, say e 1 and e 2 , then either G is the Petersen graph or the (near-bipartite) Cubeplex graph, or otherwise, each edge of G (distinct from e 1 and e 2 ) is b-invariant. As a corollary, we deduce that each essentially 4-edge-connected cubic non-near-bipartite brick G, distinct from the Petersen graph, has at least |V (G)| b-invariant edges.Proposition 3. In a cubic matching covered graph, each 3-cut is a separating cut.
A cut C := ∂(X) of a matching covered graph G is a separating cut if both its C-contractions G/X and G/X are also matching covered. A brick is solid if it is free of nontrivial separating cuts. We (Carvalho, Lucchesi and Murty) showed in [6] that the perfect matching polytope of a brick may be described without recourse to odd set constraints if and only if it is solid, and in [8] we proved that the only simple planar solid bricks are the odd wheels. The problem of characterizing nonplanar solid bricks remains unsolved.A bi-subdivision of a graph J is a graph obtained from J by replacing each of its edges by paths of odd length. A matching covered graph J is a conformal minor of a matching covered graph G if there exists a bi-subdivision H of J which is a subgraph of G such that G − V (H) has a perfect matching. For a fixed matching covered graph J, a matching covered graph G is J-based if J is a conformal minor of G and, otherwise, G is J-free. A basic result due to Lovász [14] states that every nonbipartite matching covered graph is either K 4 -based or is C 6 -based or both, where C 6 is the triangular prism. In [13], we (Kothari and Murty) showed that, for any cubic brick J, a matching covered graph G is J-free if and only if each of its bricks is J-free. We
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