Efficient reconstruction of the characteristic polynomial

The polynomial reconstruction problem, introduced by Cvetkovi\'c in 1973, asks whether the characteristic polynomial $\phi^G$ of a graph $G$ with at least $3$ vertices can be reconstructed from the polynomial deck $\{\phi^{G \setminus i}\}_{i \in V(G)}$. In this work, we prove that $\phi^G \pmod{4}$ 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 $G$, $\phi^{\overline{G}}\pmod{4}$ can be computed from $\phi^G\pmod{4}$, strengthening a recent result by Ji, Tang, Wang and Zhang. Finally, Hagos showed that the pair of characteristic polynomials $(\phi^G, \phi^{\overline{G}})$ is reconstructible from the generalized polynomial deck $\{(\phi^{G \setminus i}, \phi^{\overline{G} \setminus i})\}_{i \in V(G)}$. We also present an efficient version of this result that requires less information.