Weighted graphs defining facets: a connection between stable set and linear ordering polytopes

A graph is alpha-critical if its stability number increases whenever an edge is removed from its edge set. The class of alpha-critical graphs has several nice structural properties, most of them related to their defect which is the number of vertices minus two times the stability number. In particular, a remarkable result of Lov\'asz (1978) is the finite basis theorem for alpha-critical graphs of a fixed defect. The class of alpha-critical graphs is also of interest for at least two topics of polyhedral studies. First, Chv\'atal (1975) shows that each alpha-critical graph induces a rank inequality which is facet-defining for its stable set polytope. Investigating a weighted generalization, Lipt\'ak and Lov\'asz (2000, 2001) introduce critical facet-graphs (which again produce facet-defining inequalities for their stable set polytopes) and they establish a finite basis theorem. Second, Koppen (1995) describes a construction that delivers from any alpha-critical graph a facet-defining inequality for the linear ordering polytope. Doignon, Fiorini and Joret (2006) handle the weighted case and thus define facet-defining graphs. Here we investigate relationships between the two weighted generalizations of alpha-critical graphs. We show that facet-defining graphs (for the linear ordering polytope) are obtainable from 1-critical facet-graphs (linked with stable set polytopes). We then use this connection to derive various results on facet-defining graphs, the most prominent one being derived from Lipt\'ak and Lov\'asz's finite basis theorem for critical facet-graphs. At the end of the paper we offer an alternative proof of Lov\'asz's finite basis theorem for alpha-critical graphs

All CHSH polytopes

The correlations that admit a local hidden-variable model are described by a family of polytopes, whose facets are the Bell inequalities. The CHSH inequality is the simplest such Bell inequality and is a facet of every Bell polytope. We investigate for which Bell polytopes the CHSH inequality is also the unique (non-trivial) facet. We prove that the CHSH inequality is the unique facet for all bipartite polytopes where at least one party has a binary choice of dichotomic measurements, irrespective of the number of measurement settings and outcomes for the other party. Based on numerical results, we conjecture that it is also the unique facet for all bipartite polytopes involving two measurements per party where at least one measurement is dichotomic. Finally, we remark that these two situations can be the only ones for which the CHSH inequality is the unique facet, i.e., any polytope that does not correspond to one of these two cases necessarily has facets that are not of the CHSH form. As a byproduct of our approach, we derive a new family of facet inequalities

Determinantal Facet Ideals

We consider ideals generated by general sets of $m$-minors of an $m\times n$-matrix of indeterminates. The generators are identified with the facets of an $(m-1)$-dimensional pure simplicial complex. The ideal generated by the minors corresponding to the facets of such a complex is called a determinantal facet ideal. Given a pure simplicial complex $\Delta$, we discuss the question when the generating minors of its determinantal facet ideal $J_\Delta$ form a Gr\"obner basis and when $J_\Delta$ is a prime ideal

Dental Microwear From Natufian Hunter-Gatherers and Early Neolithic Farmers: Comparisons Within and Between Samples

Microwear patterns from Natufian hunter-gatherers (12,500–10,250 bp) and early Neolithic (10,250–7,500 bp) farmers from northern Israel are correlated with location on facet nine and related to an archaeologically suggested change in food preparation. Casts of permanent second mandibular molars are examined with a scanning electron microscope at a magnification of 500×. Digitized micrographs are taken from the upper and lower part of facet nine. Microwear patterns are recorded with an image-analysis computer program and compared within and between samples, using univariate and multivariate analyses. Comparisons within samples reveal a greater frequency of pits on the lower part of the facet among the farmers, compared to the upper part. Microwear does not vary over the facet among the hunter-gatherers. Comparisons between samples reveal larger dental pits (length and width) and wider scratches among the farmers at the bottom of the facet, compared to the hunter-gatherers. Microwear does not vary between samples at the top of the facet. The microwear patterns suggest that the Natufian to early Neolithic development led to a harder diet, and this is related to an archaeologically suggested change in food processing. The harder diet of the early farmers may have required higher bite forces that were exerted at the bottom of facet nine, in the basin of the tooth

Psychopathic traits and deviant sexual interests : the moderating role of gender

The present study examined associations between psychopathic traits and deviant sexual interests across gender in a large community sample (N = 429, 24% men). Correlation analyses supported the positive link between psychopathic traits and deviant sexual interests. Regression analyses indicated that the unique variance in the antisocial facet of psychopathy predicted all six deviant sexual interests. The interpersonal facet predicted voyeuristic and exhibitionistic interests, whereas the affective facet predicted pedophilic interests. Moderation analyses revealed that gender moderated most of the relations between the antisocial facet of psychopathy and deviant sexual interests, such that those positive associations were stronger among women. On the contrary, the associations between the interpersonal facet and voyeuristic interests, as well as between the lifestyle facet and sadistic interests, were stronger among men. Findings appear to suggest that deviant sexual interests represent a domain in which the manifestation of psychopathic traits may differ across gender. These findings emphasize the relevance of psychopathic traits for the understanding and risk assessment of sexual deviance, while suggesting the need for gender-sensitive considerations

Odd-Cycle-Free Facet Complexes and the K\"onig property

We use the definition of a simplicial cycle to define an odd-cycle-free facet complex (hypergraph). These are facet complexes that do not contain any cycles of odd length. We show that besides one class of such facet complexes, all of them satisfy the K\"onig property. This new family of complexes includes the family of balanced hypergraphs, which are known to satisfy the K\"onig property. These facet complexes are, however, not Mengerian; we give an example to demonstrate this fact.Comment: 12 pages, 11 figure

Vertex-Facet Incidences of Unbounded Polyhedra

How much of the combinatorial structure of a pointed polyhedron is contained in its vertex-facet incidences? Not too much, in general, as we demonstrate by examples. However, one can tell from the incidence data whether the polyhedron is bounded. In the case of a polyhedron that is simple and "simplicial," i.e., a d-dimensional polyhedron that has d facets through each vertex and d vertices on each facet, we derive from the structure of the vertex-facet incidence matrix that the polyhedron is necessarily bounded. In particular, this yields a characterization of those polyhedra that have circulants as vertex-facet incidence matrices.Comment: LaTeX2e, 14 pages with 4 figure