All Parity Realizable Trees

Given a graph \(G = (V, E)\) and a vertex subset \(D \subseteq V\), a subset \(S \subseteq V\) is said to realize a “parity assignment” \(D\) if for each vertex \(v \in V\) with closed neighborhood \(N[v]\) we have that \(|N[v] \cap S|\) is odd if and only if \(v \in D\). Graph \(G\) is called all parity realizable if every parity assignment \(D\) is realizable. This paper presents some examples and provides a constructive characterization of all parity realizable trees.