On rigidity, orientability and cores of random graphs with sliders

Suppose that you add rigid bars between points in the plane, and suppose that a constant fraction $q$ of the points moves freely in the whole plane; the remaining fraction is constrained to move on fixed lines called sliders. When does a giant rigid cluster emerge? Under a genericity condition, the answer only depends on the graph formed by the points (vertices) and the bars (edges). We find for the random graph $G \in \mathcal{G}(n,c/n)$ the threshold value of $c$ for the appearance of a linear-sized rigid component as a function of $q$, generalizing results of Kasiviswanathan et al. We show that this appearance of a giant component undergoes a continuous transition for $q \leq 1/2$ and a discontinuous transition for $q > 1/2$. In our proofs, we introduce a generalized notion of orientability interpolating between 1- and 2-orientability, of cores interpolating between 2-core and 3-core, and of extended cores interpolating between 2+1-core and 3+2-core; we find the precise expressions for the respective thresholds and the sizes of the different cores above the threshold. In particular, this proves a conjecture of Kasiviswanathan et al. about the size of the 3+2-core. We also derive some structural properties of rigidity with sliders (matroid and decomposition into components) which can be of independent interest.Comment: 32 pages, 1 figur

ON RIGIDITY, ORIENTABILITY AND CORES OF RANDOM GRAPHS WITH SLIDERS

International audienceSuppose that you add rigid bars between points in the plane, and suppose that a constant fraction q of the points moves freely in the whole plane; the remaining fraction is constrained to move on fixed lines called sliders. When does a giant rigid cluster emerge? Under a genericity condition, the answer only depends on the graph formed by the points (vertices) and the bars (edges). We find for the random graph G ∈ G(n, c/n) the threshold value of c for the appearance of a linear-sized rigid component as a function of q, generalizing results of [8]. We show that this appearance of a giant component undergoes a continuous transition for q ≤ 1/2 and a discontinuous transition for q > 1/2. In our proofs, we introduce a generalized notion of orientability interpolating between 1-and 2-orientability, of cores interpolating between the 2-core and the 3-core, and of extended cores interpolating between the 2 + 1-core and the 3 + 2-core; we find the precise expressions for the respective thresholds and the sizes of the different cores above the threshold. In particular, this proves a conjecture of [8] about the size of the 3 + 2-core. We also derive some structural properties of rigidity with sliders (matroid and decomposition into components) which can be of independent interest

On rigidity, orientability and cores of random graphs with sliders

Suppose that you add rigid bars between points in the plane, and suppose that a constant fraction q of the points moves freely in the whole plane; the remaining fraction is constrained to move on fixed lines called sliders. When does a giant rigid cluster emerge? Under a genericity condition, the answer only depends on the graph formed by the points (vertices) and the bars (edges). We find for the random graph G ∈ G(n, c/n) the threshold value of c for the appearance of a linear-sized rigid component as a function of q, generalizing results of [7]. We show that this appearance of a giant component undergoes a continuous transition for q ≤ 1/2 and a discontinuous transition for q > 1/2. In our proofs, we introduce a generalized notion of orientability interpolating between 1-and 2-orientability, of cores interpolating between 2-core and 3-core, and of extended cores interpolating between 2 + 1-core and 3 + 2-core; we find the precise expressions for the respective thresholds and the sizes of the different cores above the threshold. In particular, this proves a conjecture of [7] about the size of the 3 + 2-core. We also derive some structural properties of rigidity with sliders (matroid and decomposition into components) which can be of independent interest

Probability, Trees and Algorithms

The subject of this workshop were probabilistic aspects of algorithms for fundamental problems such as sorting, searching, selecting of and within data, random permutations, algorithms based on combinatorial trees or search trees, continuous limits of random trees and random graphs as well as random geometric graphs. The deeper understanding of the complexity of such algorithms and of shape characteristics of large discrete structures require probabilistic models and an asymptotic analysis of random discrete structures. The talks of this workshop focused on probabilistic, combinatorial and analytic techniques to study asymptotic properties of large random combinatorial structures

ON RIGIDITY, ORIENTABILITY AND CORES OF RANDOM GRAPHS WITH SLIDERS

International audienceSuppose that you add rigid bars between points in the plane, and suppose that a constant fraction q of the points moves freely in the whole plane; the remaining fraction is constrained to move on fixed lines called sliders. When does a giant rigid cluster emerge? Under a genericity condition, the answer only depends on the graph formed by the points (vertices) and the bars (edges). We find for the random graph G ∈ G(n, c/n) the threshold value of c for the appearance of a linear-sized rigid component as a function of q, generalizing results of [8]. We show that this appearance of a giant component undergoes a continuous transition for q ≤ 1/2 and a discontinuous transition for q > 1/2. In our proofs, we introduce a generalized notion of orientability interpolating between 1-and 2-orientability, of cores interpolating between the 2-core and the 3-core, and of extended cores interpolating between the 2 + 1-core and the 3 + 2-core; we find the precise expressions for the respective thresholds and the sizes of the different cores above the threshold. In particular, this proves a conjecture of [8] about the size of the 3 + 2-core. We also derive some structural properties of rigidity with sliders (matroid and decomposition into components) which can be of independent interest

On rigidity, orientability, and cores of random graphs with sliders

International audienc