= 0}^{v(G)} c_i(G) \lambda^{v(G)-i}$. The collection $\mathcal{PD}(G) = \{P(G-S;\lambda)\mid S\subsetneq VG\}$ is called the [*complete polynomial deck*]{} of $G$. Note that a polynomial may appear in the collection more than once. The [*rank*]{} of a graph $G$, which has $comp(G)$ components, is defined by $v(G) - comp(G)$, and its [*co-rank*]{} is defined by $e(G)-v(G) + comp(G)$. The rank polynomial of $G$ is defined by $R(G;x,y) = \sum \rho_{rs}x^ry^s$, where $\rho_{rs}$ is the number of subgraphs of $G$ with rank $r$ and co-rank $s$. The set of consecutive integers from $a$ to $b$ is denoted by $[a,b]$; in particular, $N_k = [1,k]$.

Ulam’s Conjecture
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The [*vertex deck*]{} of a graph $G$ is the collection $\mathcal{VD}(G) = \{G-v\mid v \in VG\}$, where the subgraphs in the collection are ‘unlabelled’ (or isomorphism types). Note that the vertex deck is not exactly a set: an isomorphism type may appear more than once in the vertex deck. A Graph $G$ is said to be [*reconstructible*]{} if its isomorphism class is determined by $\mathcal{VD}(G)$. Ulam [@ulam1960] proposed the following conjecture.

Graphs on more than 2 vertices are reconstructible.

A property or an invariant of a graph $G$ is said to be reconstructible if it can 