@techreport{TR-IC-03-01,
  number = {IC-03-01},
  author = {M. H. de Carvalho and C. L. Lucchesi and U. S. R. Murty},
  title = {How to Build a Brick},
  month = {February},
  year = {2003}, 
  institution = {Institute of Computing, University of Campinas},
  note = {In English, 39 pages.
    \par\selectlanguage{english}\textbf{Abstract}
       An edge $e$ of a brick $G$ is {\em removable} if $G-e$ is
       matching covered. A removable edge $e$ is {\em $b$-invariant}
       if $G-e$ has exactly one brick. A removable edge $e$ is {\em
       thin} if, for each barrier $B$ of $G-e$, the graph $G-e-B$ has
       precisely $|B|-1$ isolated vertices, each of which has degree
       two in $G-e$. Improving upon a theorem proved in [4] and [5],
       we show here that every brick different from the three basic
       bricks $K_4$, $\overline{C}_6$ and the Petersen graph has a
       $b$-invariant edge that is thin. It follows from this result
       that all bricks can be generated from the three basic bricks by
       means of four simple operations.

       A cut $C$ of a brick $G$ is a {\em separating} cut of $G$ if
       each of the two graphs obtained by shrinking a shore of $C$ to
       a single vertex is matching covered.  A brick is {\em solid} if
       it does not have any nontrivial separating cuts.  Solid bricks
       have many interesting properties ([4]) and may be thought of as
       building blocks of bricks themselves. The complexity status of
       deciding whether a given brick is solid is not known. Here, by
       using our theorem on the existence of thin edges, we show that
       every simple planar solid brick is an odd wheel.
  }
}