Efficient reconstruction of the characteristic polynomial
Thomás Jung Spier
TL;DR
This work advances the polynomial reconstruction program for graphs by (i) providing an efficient method to reconstruct the pair $(φ^G, φ^{\overline{G}})$ from the generalized polynomial deck, using walk-matrix techniques and top coefficients; (ii) proving that $φ^G$ modulo $4$ determines $φ^{\overline{G}}$ modulo $4$ in general, and that top coefficient information suffices to reconstruct modulo $4$ under natural parity and rank assumptions; (iii) deriving new infinite families of graphs for which polynomial reconstruction holds and clarifying the role of walk-matrix rank in these results; and (iv) tying these findings to homomorphism-count frameworks, highlighting broader implications for spectral graph reconstruction. The results blend walk-generating function identities with modular arithmetic to translate deck information into full or partial spectral data, yielding constructive but not always efficient algorithms and enriching the landscape of graph reconstruction questions.
Abstract
The polynomial reconstruction problem, introduced by Cvetković in 1973, asks whether the characteristic polynomial $φ^G$ of a graph $G$ with at least $3$ vertices can be reconstructed from the polynomial deck $\{φ^{G \setminus i}\}_{i \in V(G)}$. In this work, we prove that $φ^G \pmod{4}$ can be reconstructed from the polynomial deck if the number of vertices in $G$ is even or if the rank of the walk matrix of $G$ over $\mathbb{F}_2$ is less than $\lceil n/2 \rceil$. We also prove that for every graph $G$, $φ^{\overline{G}}\pmod{4}$ can be computed from $φ^G\pmod{4}$, strengthening a recent result by Ji, Tang, Wang and Zhang. Finally, Hagos showed that the pair of characteristic polynomials $(φ^G, φ^{\overline{G}})$ is reconstructible from the generalized polynomial deck $\{(φ^{G \setminus i}, φ^{\overline{G} \setminus i})\}_{i \in V(G)}$. We also present an efficient version of this result that requires less information.