We analyze local properties of sparse Erdős-Rényi graphs, where d(n)/n is the edge probability. In particular we study the behavior of very short paths. For d(n) = n o(1) we show that G(n, d(n)/n) has asymptotically almost surely (a.a.s.) bounded local treewidth and therefore is a.a.s. nowhere dense. We also discover a new and simpler proof that G(n, d/n) has a.a.s. bounded expansion for constant d. The local structure of sparse Erdős-Rényi graphs is very special: The r-neighborhood of a vertex is a tree with some additional edges, where the probability that there are m additional edges decreases with m. This implies efficient algorithms for subgraph isomorphism, in particular for finding subgraphs with small diameter. Finally, experiments suggest that preferential attachment graphs might have similar properties after deleting a small number of vertices.
In this paper, we prove that a graph G with no Ks,s-subgraph and twin-width d has r-admissibility and r-coloring numbers bounded from above by an exponential function of r and that we can construct graphs achieving such a dependency in r.
We give new decomposition theorems for classes of graphs that can be transduced in first-order logic from classes of sparse graphs -more precisely, from classes of bounded expansion and from nowhere dense classes. In both cases, the decomposition takes the form of a single colored rooted tree of bounded depth where, in addition, there can be links between nodes that are not related in the tree. The constraint is that the structure formed by the tree and the links has to be sparse. Using the decomposition theorem for transductions of nowhere dense classes, we show that they admit low-shrubdepth covers of size O(n ε ), where n is the vertex count and ε > 0 is any fixed real. This solves an open problem posed by Gajarský et al. (ACM TOCL '20) and also by Briański et al. (SIDMA '21). * This work is a part of projects LIPA (JG, SK) and BOBR (JG, MP, SzT) that have received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreements No. 683080 and 948057, respectively).
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