On giant components and treewidth in the layers model

Abstract

ABSTRACT:Given an undirected n-vertex graph G(V , E) and an integer k, let T k (G) denote the random vertex induced subgraph of G generated by ordering V according to a random permutation π and including in T k (G) those vertices with at most k −1 of their neighbors preceding them in this order. The distribution of subgraphs sampled in this manner is called the layers model with parameter k. The layers model has found applications in studying -degenerate subgraphs, the design of algorithms for the maximum independent set problem, and in bootstrap percolation.In the current work we expand the study of structural properties of the layers model. We prove that there are 3-regular graphs G for which with high probability T 3 (G) has a connected component of size (n), and moreover, T 3 (G) has treewidth (n). In contrast, T 2 (G) is known to be a forest (hence of treewidth 1), and we prove that if G is of bounded degree then with high probability the largest connected component in T 2 (G) is of size O(log n). We also consider the infinite grid Z 2 , for which we prove that T 4 (Z 2 ) contains a unique infinite connected component with probability 1.

Infinite and Giant Components in the Layers Percolation Model

“…Construct now an auxiliary (random) multigraph H with two sets of vertices, U 1 and U 2 . Every component in L serves as a vertex in U 1 , and the number of good D vertices in a component serves as the number of half-edges of the corresponding U 1 -vertex (to avoid confusion we shall refer to the vertices in U 1 as super-vertices (following the terminology of [4]). Hence, on D(c ′ n) ∩ F , we have that c ′ n/a k ≤ |U 1 | ≤ n/a k and as every good vertex in D within a component in L contributes one edge to the degree of the super-vertex of H corresponding to its component in T 3 (G ′ ), we get that u∈U 1 d u ≥ c ′ n.…”

“…To determine that H is likely to have a giant component we use Theorem 5.1. We first need the following proposition taken from [4]. Proposition 5.3.…”

“…In Section 3 we prove an analogous result in the infinite setup, while dropping the assumption that G is of bounded degree. The only infinite graph considered in [4] is Z 2 , for which it was shown that T 4 (Z 2 ) a.s. has a unique infinite connected component (which we also call an infinite cluster ).…”

“…Theorem 4.1 asserts that T 4 (Z d ) contains an open infinite monotone path 2 a.s., for all sufficiently large d. We expect Theorem 3 to hold for all d, however it seems that even with a more careful analysis, the argument in the proof of Theorem 3 cannot be used to prove this, because of the restriction of the argument to monotone paths. In [4] it was shown that for random 3-regular graphs, w.h.p. 3 T 3 contains a giant component 4 .…”

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In this work we continue the investigation launched in [4] of the structural properties of the structural properties of the Layers model, a dependent percolation model. Given an undirected graph G = (V, E) and an integer k, let T k (G) denote the random vertexinduced subgraph of G, generated by ordering V according to Uniform[0, 1] i.i.d.

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