@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.
  }
}