The unpublished 1990 technical report reproduced here introduced the $t$-reinforced $k$-fence inequalities, the first facets of the acyclic subgraph polytope that are not rank inequalities: each simple $k$-fence induces
$k-2$ of them, one for each $1\le t\le k-2$. The report proved this with the spectral method, using a coherent configuration and computer algebra. It went unpublished because referees preferred conventional proofs! Here, we reproduce the report with minor corrections, but otherwise with the same appearance. We then develop the spectral method further. We extend Theorem 4.4 of \href{https://doi.org/10.1016/0024-3795(91)90074-7}{Lee and Lee (1991)} to concordant polytopes that are not balanced, compute the completely reduced form of the fence configuration, and obtain a short proof of the facet result that needs no symbolic computation. We determine the dimension and a maximal irredundant equation system of
every face of the acyclic subgraph polytope of a simple fence that shares its symmetry, and show that the symmetric facets are exactly the $t$-reinforced inequalities. Finally, we show that each block rank of the completely reduced form is the dimension of the span of a few explicit statistics, and use this to determine the symmetric faces of the linear ordering polytope and to treat an unbalanced clique face of the forest polytope.
Reinforcing Old Fences Gives New Facets and More
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