@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. } }