deck. Here we present basic facts about the characteristic polynomial, and outline the idea of Section \[sec-poly\].

A graph is called [*elementary*]{} if each of its components is $1$-regular or $2$-regular. In other words, each component of an elementary graph is a single edge ($K_2$) or a cycle ($C_r; r >2$).

Let $L_i$ be the collection of all unlabelled $i$-vertex elementary graphs. So, $L_0 = \{\Phi \}$, $L_1 = \emptyset$, $L_2 = \{K_2\}$, and so on.

\[lem-sachs\] (Proposition 7.3 in [@biggs1993]) Coefficients of the characteristic polynomial of a graph $G$ are given by $$(-1)^i c_i(G) = \sum_{F \in L_i, F\subseteq G} (-1)^{r(F)} 2^{s(F)}$$ where $r(F)$ and $s(F)$ are the rank and the co-rank of $F$, respectively.

Thus, $c_0(G) = 1$, $c_1(G) = 0$, and $c_2(G) = e(G)$.

(Note 2d in [@biggs1993]) \[lem-derivative\] Let $P'(G;\lambda)$ denote the first derivative of $P(G;\lambda)$ with respect to $\lambda$. Then, $$\label{eq-derivative}
P'(G;\lambda) = \sum_{u\in VG}P(G-u;\lambda)$$

From the above two lemmas, it is clear that the problem of reconstructing a characteristic polynomial (either from the vertex deck or the complete polynomial deck) reduces to computing the coefficient $c_{v(G)}(G)$, which is the constant term in $P(G;\lambda)$. This in turn is a problem of counting the elementary spanning 