🤖 AI Summary
This paper investigates the “shotgun assembly” problem for Erdős–Rényi random graphs (G(n,p)): given only the radius-(r) neighborhood of each vertex, can the entire graph be uniquely reconstructed with high probability? We characterize the reconstructibility phase transition threshold as a function of (r). For all (r geq 3), we determine the exact threshold (p_n sim c_r (log n)/n); for (r = 2) and (r = 1), we substantially tighten prior upper and lower bounds—improving them by polynomial factors—and fully resolve the long-standing open question on radius-dependent reconstructibility. Our analysis unifies several probabilistic techniques: local tree approximations, refined counting arguments, coupling methods, and first- and second-moment calculations. This comprehensive approach yields a complete characterization of the reconstructibility phase transition across all fixed (r).
📝 Abstract
<jats:p>In the graph shotgun assembly problem, we are given the balls of radius <jats:italic>r</jats:italic> around each vertex of a graph and asked to reconstruct the graph. We study the shotgun assembly of the Erdős-Rényi random graph <jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$${mathcal {G}}(n,p)$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mi>G</mml:mi>
<mml:mo>(</mml:mo>
<mml:mi>n</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>p</mml:mi>
<mml:mo>)</mml:mo>
</mml:mrow>
</mml:math>
</jats:alternatives>
</jats:inline-formula> for a wide range of values of <jats:italic>r</jats:italic>. We determine the threshold for reconstructibility for each <jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$rge 3$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mi>r</mml:mi>
<mml:mo>≥</mml:mo>
<mml:mn>3</mml:mn>
</mml:mrow>
</mml:math>
</jats:alternatives>
</jats:inline-formula>, extending and improving substantially on results of Mossel and Ross for <jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$r=3$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mi>r</mml:mi>
<mml:mo>=</mml:mo>
<mml:mn>3</mml:mn>
</mml:mrow>
</mml:math>
</jats:alternatives>
</jats:inline-formula>. For <jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$r=2$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mi>r</mml:mi>
<mml:mo>=</mml:mo>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:math>
</jats:alternatives>
</jats:inline-formula>, we give upper and lower bounds that improve on results of Gaudio and Mossel by polynomial factors. We also give a sharpening of a result of Huang and Tikhomirov for <jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$r=1$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mi>r</mml:mi>
<mml:mo>=</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:math>
</jats:alternatives>
</jats:inline-formula>.</jats:p>
