The polynomial reconstruction problem for graphs having cut-vertices of degree two

In the polynomial reconstruction problem (PRP), the characteristic polynomial $\varphi (G,x)$ of a graph $G$ is sought from the polynomial deck $\mathrm{PD}\left(G\right)$ containing the characteristic polynomials of the $n$ vertex-deleted subgraphs of $G$. The diagonal entries of the adjugate matrix $adj(G,x)$ of $G$ are the elements of $\mathrm{PD}\left(G\right)$. The PRP is not completely solved for graphs having vertices of degree one. In this paper, we use $adj(G,x)$ to successfully obtain $\varphi (G,x)$ from $\mathrm{PD}\left(G\right)$ for certain graphs having a vertex of degree one whose characteristic polynomial is not discoverable using results from current literature. Our methods require such graphs to have a vertex of degree one adjacent to a vertex of degree two, and this latter vertex would then be a cut-vertex of $G$. We thus extend this idea to partially solve the PRP for more general graphs that have a cut-vertex of degree two which is not necessarily adjacent to vertices of degree one, presenting an algorithm that provides $\varphi (G,x)$ from $\mathrm{PD}\left(G\right)$ when $G$ has such a cut-vertex.