Semi-Random Graphs, Robust Asymmetry, and Reconstruction

Abstract

The Graph Reconstruction Conjecture famously posits that any undirected graph on at least three vertices is determined up to isomorphism by its family of (unlabeled) induced subgraphs. At present, the conjecture admits partial resolutions of two types: 1) casework-based demonstrations of reconstructibility for families of graphs satisfying certain structural properties, and 2) probabilistic arguments establishing reconstructibility of random graphs by leveraging average-case phenomena. While results in the first category capture the worst-case nature of the conjecture, they play a limited role in understanding the general case. Results in the second category address much larger graph families, but it remains unclear how heavily the necessary arguments rely on optimistic distributional properties. Drawing on the perspectives of smoothed and semi-random analysis, we study the robustness of what are arguably the two most fundamental properties in this latter line of work: asymmetry and uniqueness of subgraphs. Notably, we find that various semi-random graph distributions exhibit these properties asymptotically, much like their Erd\H{o}s-R\'enyi counterparts. In particular, Bollob\'as (1990) demonstrated that almost all Erd\H{o}s-R\'enyi random graphs $G = (V, E) \sim \mathscr{G}(n, p)$ enjoy the property that their induced subgraphs on $n - \Theta(1)$ vertices are asymmetric and mutually non-isomorphic, for $1 - p, p = \Omega(\log(n) / n)$. We show that this property is robust against perturbation -- even when an adversary is permitted to add/remove each vertex pair in $V^{(2)}$ with (independent) arbitrarily large constant probability. Exploiting this result, we derive asymptotic characterizations of asymmetry in random graphs with planted structure and bounded adversarial corruptions, along with improved bounds on the probability mass of nonreconstructible graphs in $\mathscr{G}(n, p)$.