Abstract
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.
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Also known as principal vertices.
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Dreier, J., Kuinke, P., Xuan, B.L., Rossmanith, P. (2018). Local Structure Theorems for Erdős–Rényi Graphs and Their Algorithmic Applications. In: Tjoa, A., Bellatreche, L., Biffl, S., van Leeuwen, J., Wiedermann, J. (eds) SOFSEM 2018: Theory and Practice of Computer Science. SOFSEM 2018. Lecture Notes in Computer Science(), vol 10706. Edizioni della Normale, Cham. https://doi.org/10.1007/978-3-319-73117-9_9
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