Perfectly contractile graphs and quadratic toric rings

Perfect graphs form one of the distinguished classes of finite simple graphs. In 2006, Chudnovsky, Robertson, Saymour and Thomas proved that a graph is perfect if and only if it has no odd holes and no odd antiholes as induced subgraphs, which was conjectured by Berge. We consider the class ${\mathcal A}$ of graphs that have no odd holes, no antiholes and no odd stretchers as induced subgraphs. In particular, every graph belonging to ${\mathcal A}$ is perfect. Everett and Reed conjectured that a graph belongs to ${\mathcal A}$ if and only if it is perfectly contractile. In the present paper, we discuss graphs belonging to ${\mathcal A}$ from a viewpoint of commutative algebra. In fact, we conjecture that a perfect graph $G$ belongs to ${\mathcal A}$ if and only if the toric ideal of the stable set polytope of $G$ is generated by quadratic binomials. Especially, we show that this conjecture is true for Meyniel graphs, perfectly orderable graphs, and clique separable graphs, which are perfectly contractile graphs.Comment: 10 page

Precoloring co-Meyniel graphs

The pre-coloring extension problem consists, given a graph $G$ and a subset of nodes to which some colors are already assigned, in finding a coloring of $G$ with the minimum number of colors which respects the pre-coloring assignment. This can be reduced to the usual coloring problem on a certain contracted graph. We prove that pre-coloring extension is polynomial for complements of Meyniel graphs. We answer a question of Hujter and Tuza by showing that ``PrExt perfect'' graphs are exactly the co-Meyniel graphs, which also generalizes results of Hujter and Tuza and of Hertz. Moreover we show that, given a co-Meyniel graph, the corresponding contracted graph belongs to a restricted class of perfect graphs (``co-Artemis'' graphs, which are ``co-perfectly contractile'' graphs), whose perfectness is easier to establish than the strong perfect graph theorem. However, the polynomiality of our algorithm still depends on the ellipsoid method for coloring perfect graphs

Coloring Artemis graphs

We consider the class A of graphs that contain no odd hole, no antihole, and no ``prism'' (a graph consisting of two disjoint triangles with three disjoint paths between them). We show that the coloring algorithm found by the second and fourth author can be implemented in time O(n^2m) for any graph in A with n vertices and m edges, thereby improving on the complexity proposed in the original paper

Perfect Graphs

This chapter is a survey on perfect graphs with an algorithmic flavor. Our emphasis is on important classes of perfect graphs for which there are fast and efficient recognition and optimization algorithms. The classes of graphs we discuss in this chapter are chordal, comparability, interval, perfectly orderable, weakly chordal, perfectly contractile, and chi-bound graphs. For each of these classes, when appropriate, we discuss the complexity of the recognition algorithm and algorithms for finding a minimum coloring, and a largest clique in the graph and its complement

Bulk rheology and microrheology of active fluids

We simulate macroscopic shear experiments in active nematics and compare them with microrheology simulations where a spherical probe particle is dragged through an active fluid. In both cases we define an effective viscosity: in the case of bulk shear simulations this is the ratio between shear stress and shear rate, whereas in the microrheology case it involves the ratio between the friction coefficient and the particle size. We show that this effective viscosity, rather than being solely a property of the active fluid, is affected by the way chosen to measure it, and strongly depends on details such as the anchoring conditions at the probe surface and on both the system size and the size of the probe particle.Comment: 12 pages, 10 figure