Relevance of visco-plastic theory in a multi-directional inhomogeneous granular flow

We confront a recent visco-plastic description of dense granular flows [P. Jop et al, Nature, {\bf 441} (2006) 727] with multi-directional inhomogeneous steady flows observed in non-smooth contact dynamics simulations of 2D half-filled rotating drums. Special attention is paid to check separately the two underlying fundamental statements into which the considered theory can be recast, namely (i) a single relation between the invariants of stress and strain rate tensors and (ii) the alignment between these tensors. Interestingly, the first prediction is fairly well verified over more than four decades of small strain rate, from the surface rapid flow to the quasi-static creep phase, where it is usually believed to fail because of jamming. On the other hand, the alignment between stress and strain rate tensors is shown to fail over the whole flow, what yields an apparent violation of the visco-plastic rheology when applied without care. In the quasi-static phase, the particularly large misalignment is conjectured to be related to transient dilatancy effects

Reconfiguring Independent Sets in Claw-Free Graphs

We present a polynomial-time algorithm that, given two independent sets in a claw-free graph $G$, decides whether one can be transformed into the other by a sequence of elementary steps. Each elementary step is to remove a vertex $v$ from the current independent set $S$ and to add a new vertex $w$ (not in $S$) such that the result is again an independent set. We also consider the more restricted model where $v$ and $w$ have to be adjacent

Diphasic non-local model for granular surface flows

Considering recent results revealing the existence of multi-scale rigid clusters of grains embedded in granular surface flows, i.e. flows down an erodible bed, we describe here the surface flows rheology through a non-local constitutive law. The predictions of the resulting model are compared quantitatively to experimental results: The model succeeds to account for the counter-intuitive shape of the velocity profile observed in experiments, i.e. a velocity profile decreasing exponentially with depth in the static phase and remaining linear in the flowing layer with a velocity gradient independent of both the flowing layer thickness, the angle between the flow and the horizontal, and the coefficient of restitution of the grains. Moreover, the scalings observed in rotating drums are recovered, at least for small rotating speed.Comment: 7 pages, submitted to Europhys. Let

Experimental study of granular surface flows via a fast camera: a continuous description

Depth averaged conservation equations are written for granular surface flows. Their application to the study of steady surface flows in a rotating drum allows to find experimentally the constitutive relations needed to close these equations from measurements of the velocity profile in the flowing layer at the center of the drum and from the flowing layer thickness and the static/flowing boundary profiles. The velocity varies linearly with depth, with a gradient independent of both the flowing layer thickness and the static/flowing boundary local slope. The first two closure relations relating the flow rate and the momentum flux to the flowing layer thickness and the slope are then deduced. Measurements of the profile of the flowing layer thickness and the static/flowing boundary in the whole drum explicitly give the last relation concerning the force acting on the flowing layer. Finally, these closure relations are compared to existing continuous models of surface flows.Comment: 20 pages, 11 figures, submitted to Phys. FLuid

Fixed-Parameter Tractability of Token Jumping on Planar Graphs

Suppose that we are given two independent sets $I_0$ and $I_r$ of a graph such that $|I_0| = |I_r|$, and imagine that a token is placed on each vertex in $I_0$. The token jumping problem is to determine whether there exists a sequence of independent sets which transforms $I_0$ into $I_r$ so that each independent set in the sequence results from the previous one by moving exactly one token to another vertex. This problem is known to be PSPACE-complete even for planar graphs of maximum degree three, and W[1]-hard for general graphs when parameterized by the number of tokens. In this paper, we present a fixed-parameter algorithm for the token jumping problem on planar graphs, where the parameter is only the number of tokens. Furthermore, the algorithm can be modified so that it finds a shortest sequence for a yes-instance. The same scheme of the algorithms can be applied to a wider class of graphs, $K_{3,t}$-free graphs for any fixed integer $t \ge 3$, and it yields fixed-parameter algorithms

A reconfigurations analogue of Brooks’ theorem.

Let G be a simple undirected graph on n vertices with maximum degree Δ. Brooks’ Theorem states that G has a Δ-colouring unless G is a complete graph, or a cycle with an odd number of vertices. To recolour G is to obtain a new proper colouring by changing the colour of one vertex. We show that from a k-colouring, k > Δ, a Δ-colouring of G can be obtained by a sequence of O(n 2) recolourings using only the original k colours unless G is a complete graph or a cycle with an odd number of vertices, or k = Δ + 1, G is Δ-regular and, for each vertex v in G, no two neighbours of v are coloured alike. We use this result to study the reconfiguration graph R k (G) of the k-colourings of G. The vertex set of R k (G) is the set of all possible k-colourings of G and two colourings are adjacent if they differ on exactly one vertex. It is known that if k ≤ Δ(G), then R k (G) might not be connected and it is possible that its connected components have superpolynomial diameter, if k ≥ Δ(G) + 2, then R k (G) is connected and has diameter O(n 2). We complete this structural classification by settling the missing case: if k = Δ(G) + 1, then R k (G) consists of isolated vertices and at most one further component which has diameter O(n 2). We also describe completely the computational complexity classification of the problem of deciding whether two k-colourings of a graph G of maximum degree Δ belong to the same component of R k (G) by settling the case k = Δ(G) + 1. The problem is O(n 2) time solvable for k = 3, PSPACE-complete for 4 ≤ k ≤ Δ(G), O(n) time solvable for k = Δ(G) + 1, O(1) time solvable for k ≥ Δ(G) + 2 (the answer is always yes)

Block to granular-like transition in dense bubble flows

We have experimentally investigated 2-dimensional dense bubble flows underneath inclined planes. Velocity profiles and velocity fluctuations have been measured. A broad second-order phase transition between two dynamical regimes is observed as a function of the tilt angle $\theta$. For low $\theta$ values, a block motion is observed. For high $\theta$ values, the velocity profile becomes curved and a shear velocity gradient appears in the flow.Comment: Europhys. Lett. (2003) in pres

Token Jumping in minor-closed classes

Given two $k$-independent sets $I$ and $J$ of a graph $G$, one can ask if it is possible to transform the one into the other in such a way that, at any step, we replace one vertex of the current independent set by another while keeping the property of being independent. Deciding this problem, known as the Token Jumping (TJ) reconfiguration problem, is PSPACE-complete even on planar graphs. Ito et al. proved in 2014 that the problem is FPT parameterized by $k$ if the input graph is $K_{3,\ell}$-free. We prove that the result of Ito et al. can be extended to any $K_{\ell,\ell}$-free graphs. In other words, if $G$ is a $K_{\ell,\ell}$-free graph, then it is possible to decide in FPT-time if $I$ can be transformed into $J$. As a by product, the TJ-reconfiguration problem is FPT in many well-known classes of graphs such as any minor-free class

Colouring triangle-free graphs with local list sizes

We prove two distinct and natural refinements of a recent breakthrough result of Molloy (and a follow-up work of Bernshteyn) on the (list) chromatic number of triangle-free graphs. In both our results, we permit the amount of colour made available to vertices of lower degree to be accordingly lower. One result concerns list colouring and correspondence colouring, while the other concerns fractional colouring. Our proof of the second illustrates the use of the hard-core model to prove a Johansson-type result, which may be of independent interest.Comment: 16 pages; v2 includes minor corrections after review; to appear in Random Structures & Algorithm

Independent Set Reconfiguration in Cographs

We study the following independent set reconfiguration problem, called TAR-Reachability: given two independent sets $I$ and $J$ of a graph $G$, both of size at least $k$, is it possible to transform $I$ into $J$ by adding and removing vertices one-by-one, while maintaining an independent set of size at least $k$ throughout? This problem is known to be PSPACE-hard in general. For the case that $G$ is a cograph (i.e. $P_4$-free graph) on $n$ vertices, we show that it can be solved in time $O(n^2)$, and that the length of a shortest reconfiguration sequence from $I$ to $J$ is bounded by $4n-2k$, if such a sequence exists. More generally, we show that if $X$ is a graph class for which (i) TAR-Reachability can be solved efficiently, (ii) maximum independent sets can be computed efficiently, and which satisfies a certain additional property, then the problem can be solved efficiently for any graph that can be obtained from a collection of graphs in $X$ using disjoint union and complete join operations. Chordal graphs are given as an example of such a class $X$