{"paper":{"title":"Planting trees in graphs, and finding them back","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Don Towsley, Laurent Massouli\\'e, Ludovic Stephan","submitted_at":"2018-11-05T15:37:31Z","abstract_excerpt":"In this paper we study detection and reconstruction of planted structures in Erd\\H{o}s-R\\'enyi random graphs. Motivated by a problem of communication security, we focus on planted structures that consist in a tree graph. For planted line graphs, we establish the following phase diagram. In a low density region where the average degree $\\lambda$ of the initial graph is below some critical value $\\lambda_c=1$, detection and reconstruction go from impossible to easy as the line length $K$ crosses some critical value $f(\\lambda)\\ln(n)$, where $n$ is the number of nodes in the graph. In the high de"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.01800","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}