On the dominating set polytope of web graphs

The set covering problem on circulant matrices: polynomial instances and the relation with the dominating set problem on webs

“…Argiroffo and Bianchi obtain in [2] a family of facets defining inequalities for Q(C k n ) associated with circular minors corresponding to d = 1 according to the notation in Theorem 2.1. This result was generalized for every circulant minor as it is stated in the following Theorem 2.2 [5] Let C k n be a minor of C k n and for j = 1 . .…”

Circuits and Circulant Minors

“…Working on the set covering problem on circular matrices [2], we have found a sufficient condition for a circular matrix to have a circulant minor, also expressed in terms of circuits in the following digraph associated with the matrix:…”

“…In the following we denote by p the winding number of Γ and assume that the circuit has s essential bullets [2] for further details). Then, denoting by v j the vertex b j + t j − 1, we define the block B j := [b j , v j ] n which can be a circle block, a cross block or a bullet block, depending on the vertex class that b j + 1 belongs to.…”

“…The inequalities involved in this formulation are related to directed circuits in another auxiliary digraph associated with the matrix. More recently, we have obtained a perfect formulation for SCP [2]. Once again, directed circuits in a certain digraph are related to the inequalities that appear in the linear description.…”

“…In [2] we also stated a necessary condition for a circular matrix to have a circulant minor. In this paper we further develop this result and completely characterize circulant minors of circular matrices in terms of directed circuits in its associated digraph.…”