Brick Generation and Conformal Subgraphs

Abstract

A nontrivial connected graph is matching covered if each of its edges lies in a perfect matching. Two types of decompositions of matching covered graphs, namely ear decompositions and tight cut decompositions, have played key roles in the theory of these graphs. Any tight cut decomposition of a matching covered graph results in an essentially unique list of special matching covered graphs, called bricks (which are nonbipartite and 3-connected) and braces (which are bipartite).

A fundamental theorem of LovU+00E1sz (1983) states that every nonbipartite matching covered graph admits an ear decomposition starting with a bi-subdivision of $K_4$ or of the triangular prism $\overline{C_6}$. This led Carvalho, Lucchesi and Murty (2003) to pose two problems: (i) characterize those nonbipartite matching covered graphs which admit an ear decomposition starting with a bi-subdivision of $K_4$, and likewise, (ii) characterize those which admit an ear decomposition starting with a bi-subdivision of $\overline{C_6}$.

In the first part of this thesis, we solve these problems for the special case of planar graphs. In Chapter 2, we reduce these problems to the case of bricks, and in Chapter 3, we solve both problems when the graph under consideration is a planar brick.

A nonbipartite matching covered graph G is near-bipartite if it has a pair of edges U+03B1 and U+03B2 such that G-{U+03B1,U+03B2} is bipartite and matching covered; examples are $K_4$ and $\overline{C_6}$. The first nonbipartite graph in any ear decomposition of a nonbipartite graph is a bi-subdivision of a near-bipartite graph. For this reason, near-bipartite graphs play a central role in the theory of matching covered graphs. In the second part of this thesis, we establish generation theorems which are specific to near-bipartite bricks.

Deleting an edge e from a brick G results in a graph with zero, one or two vertices of degree two, as G is 3-connected. The bicontraction of a vertex of degree two consists of contracting the two edges incident with it; and the retract of G-e is the graph J obtained from it by bicontracting all its vertices of degree two. The edge e is thin if J is also a brick. Carvalho, Lucchesi and Murty (2006) showed that every brick, distinct from $K_4$, $\overline{C_6}$ and the Petersen graph, has a thin edge.

In general, given a near-bipartite brick G and a thin edge e, the retract J of G-e need not be near-bipartite. In Chapter 5, we show that every near-bipartite brick G, distinct from $K_4$ and $\overline{C_6}$, has a thin edge e such that the retract J of G-e is also near-bipartite. Our theorem is a refinement of the result of Carvalho, Lucchesi and Murty which is appropriate for the restricted class of near-bipartite bricks.

For a simple brick G and a thin edge e, the retract of G-e may not be simple. It was established by Norine and Thomas (2007) that each simple brick, which is not in any of five well-defined infinite families of graphs, and is not isomorphic to the Petersen graph, has a thin edge such that the retract J of G-e is also simple.

In Chapter 6, using our result from Chapter 5, we show that every simple near-bipartite brick G has a thin edge e such that the retract J of G-e is also simple and near-bipartite, unless G belongs to any of eight infinite families of graphs. This is a refinement of the theorem of Norine and Thomas which is appropriate for the restricted class of near-bipartite bricks.