# Whitney Complexes and the Subideals of Monomial Ideals

Reconstruction of a graph is whether one can uniquely determine a graph from the set of unlabeled subgraphs obtained by deleting a vertex (vertex deck) or the set of unlabeled subgraphs obtained by deleting an edge (edge-deck).

In 1964, Harary conjectured any two graphs with at least four edges and the same edge-deck are isomorphic. Later Hemminger proved that the edge reconstruction conjecture for graphs is equivalent to the vertex reconstruction conjecture for line graphs using some theorem proved by Whitney in 1932. He had proved that if the line graphs of two simple graph $G$ and $H$ are isomorphic, then $G$ and $H$ are also isomorphic except for the case $K_3$ and $K_{1,3}$.

Trying to extend Whitney's theorem to hypergraphs, Berge introduced two hypergraphs $\mathcal{E}_p$ and $\mathcal{O}_p$ and proved that if two haypergraphs have isomorphic p-edge-deck then they are isomorphic only if they do not contain an $\mathcal{E}_p-\mathcal{O}_p$ pair. This result was improved by Gardner in 1987.

Considering the strong combinatorial binding between $\mathcal{E}_p$ and $\mathcal{O}_p$, one expects strong similarities between their algebraic properties. In this research we mainly focus on studying and discovering these properties.