@techreport{TR-IC-08-16,
number = {IC-08-16},
author = {Marcelo H. de Carvalho and
Cláudio L. Lucchesi and
U. S. R. Murty},
title = {Generating Simple Bricks and Braces},
month = {July},
year = {2008},
institution = {Institute of Computing, University of Campinas},
note = {In English, 38 pages.
\par\selectlanguage{english}\textbf{Abstract}
The \emph{bicontraction} of a vertex of degree two in a graph
consists of contracting both the edges incident with that
vertex. The \emph{retract} of a graph is the graph obtained
from it by bicontracting all its vertices of degree two. An
edge $e$ of a brick $G$ is \emph{thin} if the retract of $G-e$
is a brick, and is \emph{strictly thin} if that retract is a
simple brick. Thin and strictly thin edges in braces on six or
more vertices are similarly defined. We showed in a previous
paper that every brick distinct from $K_4$, $\overline{C_6}$
and the Petersen graph has a thin edge. In the first part of
this paper we show that every brace has a thin edge.
\par McCuaig showed that every brace, which is not a biwheel
or a prism or a Möbius ladder, has a strictly thin
edge. Analogously, Norine and Thomas showed that every brick,
which is different from the Petersen graph and is not in any
one of five well-defined infinite families of graphs, has a
strictly thin edge. These theorems yield procedures for
generating simple braces and bricks, respectively. In the
second part of the paper we show that the results of McCuaig
on braces, and of Norine and Thomas on bricks, may be deduced
fairly easily from ours mentioned above. The proofs of these
results are so remarkably similar that we are able to present
them simultaneously.
}
}