On dominating set polyhedra of circular interval graphs

Bianchi, Silvia; Nasini, Graciela L.; Tolomei, Paola B.; Torres, Luis Miguel

doi:10.1016/j.disc.2020.112283

Cited by 2 publications

(9 citation statements)

References 28 publications

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“…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: Definition 2.1. [2] Given a circular matrix A, let F (A) be the directed graph whose set of vertices is [n] and whose arcs are of the form ( i − 1, u i ), for every i ∈ [m] (called row arcs) and (j, j − 1), (j − 1, j) with j ∈ [n] (termed as reverse short arcs and forward short arcs, respectively).…”

Section: Notations and Preliminary Resultsmentioning

confidence: 99%

“…In the following we denote by p the winding number of Γ and assume that the circuit has s essential bullets {b [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.…”

Section: Notations and Preliminary Resultsmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

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Circuits and Circulant Minors

Bianchi

Nasini

Tolomei

et al. 2019

Electronic Notes in Theoretical Computer Science

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Circulant contraction minors play a key role for characterizing ideal circular matrices in terms of minimally non ideal structures. In this article we prove necessary and sufficient conditions for a circular matrix A to have circulant contraction minors in terms of circuits in a digraph associated with A. In the particular case when A itself is a circulant matrix, our result provides an alternative characterization to the one previously known from the literature.

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Section: Notations and Preliminary Resultsmentioning

confidence: 99%

Section: Notations and Preliminary Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Circuits and Circulant Minors

Bianchi

Nasini

Tolomei

et al. 2019

Electronic Notes in Theoretical Computer Science

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“…N.B. Our techniques and theorems could be extended naturally to the set covering problem in circulant matrices (see Bianchi et al 2017 ): in fact, row family inequalities are the counterpart of clique family inequalities in this setting and the stable set polytope of circulant graphs and the set covering polytope of circulant matrices bear strong resemblance. Observe now that trivially the minimum size of a root of C F I (Γ ) is equal to β(Γ ); the first statement of the lemma follows then from Lemma 27.…”

Section: A Concluding Remarkmentioning

confidence: 99%

On the facets of stable set polytopes of circular interval graphs

Oriolo

Stauffer

2022

Ann Oper Res

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A linear description of the stable set polytope STAB(G) of a quasi-line graph G is given in Eisenbrand et al. (Combinatorica 28(1):45–67, 2008), where the so called Ben Rebea Theorem (Oriolo in Discrete Appl Math 132(3):185–201, 2003) is proved. Such a theorem establishes that, for quasi-line graphs, STAB(G) is completely described by non-negativity constraints, clique inequalities, and clique family inequalities (CFIs). As quasi-line graphs are a superclass of line graphs, Ben Rebea Theorem can be seen as a generalization of Edmonds’ characterization of the matching polytope (Edmonds in J Res Natl Bureau Stand B 69:125–130, 1965), showing that the matching polytope can be described by non-negativity constraints, degree constraints and odd-set inequalities. Unfortunately, the description given by the Ben Rebea Theorem is not minimal, i.e., it is not known which are the (non-rank) clique family inequalities that are facet defining for STAB(G). To the contrary, it would be highly desirable to have a minimal description of STAB(G), pairing that of Edmonds and Pulleyblank (in: Berge, Chuadhuri (eds) Hypergraph seminar, pp 214–242, 1974) for the matching polytope. In this paper, we start the investigation of a minimal linear description for the stable set polytope of quasi-line graphs. We focus on circular interval graphs, a subclass of quasi-line graphs that is central in the proof of the Ben Rebea Theorem. For this class of graphs, we move an important step forward, showing some strong sufficient conditions for a CFI to induce a facet of STAB(G). In particular, such conditions come out to be related to the existence of certain proper circulant graphs as subgraphs of G. These results allows us to settle two conjectures on the structure of facet defining inequalities of the stable set polytope of circulant graphs (Pêcher and Wagler in Math Program 105:311–328, 2006) and of (fuzzy) circular graphs (Oriolo and Stauffer in Math Program 115:291–317, 2008), and to slightly refine the Ben Rebea Theorem itself.

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On dominating set polyhedra of circular interval graphs

Cited by 2 publications

References 28 publications

Circuits and Circulant Minors

Circuits and Circulant Minors

On the facets of stable set polytopes of circular interval graphs

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