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 3connected) 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 bisubdivision 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 bisubdivision of $K_4$, and likewise, (ii) characterize those which admit an ear decomposition starting with a bisubdivision 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 nearbipartite 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 bisubdivision of a nearbipartite graph. For this reason, nearbipartite 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 nearbipartite 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 3connected. The bicontraction of a vertex of degree two consists of contracting the two edges incident with it; and the retract of Ge 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 nearbipartite brick G and a thin edge e, the retract J of Ge need not be nearbipartite. In Chapter 5, we show that every nearbipartite brick G, distinct from $K_4$ and $\overline{C_6}$, has a thin edge e such that the retract J of Ge is also nearbipartite. Our theorem is a refinement of the result of Carvalho, Lucchesi and Murty which is appropriate for the restricted class of nearbipartite bricks.
For a simple brick G and a thin edge e, the retract of Ge may not be simple. It was established by Norine and Thomas (2007)
that each simple brick, which is not in any of five welldefined infinite families of graphs, and is not isomorphic to the Petersen graph, has a thin edge such that the retract J of Ge is also simple.
In Chapter 6, using our result from Chapter 5, we show that every simple nearbipartite brick G has a thin edge e such that the retract J of Ge is also simple and nearbipartite, 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 nearbipartite bricks.
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Cite this version of the work
Nishad Kothari
(2016).
Brick Generation and Conformal Subgraphs. UWSpace.
http://hdl.handle.net/10012/10376
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