An extremal result for geometries in the one-way measurement model

We present an extremal result for the class of graphs G which (together with some specified sets of input and output vertices, I and O) have a certain "flow" property introduced by Danos and Kashefi for the one-way measurement model of quantum computation. The existence of a flow for a triple (G,I,O) allows a unitary embedding to be derived from any choice of measurement bases allowed in the one-way measurement model. We prove an upper bound on the number of edges that a graph G may have, in order for a triple (G,I,O) to have a flow for some $I, O \subseteq V(G)$, in terms of the number of vertices in G and O. This implies that finding a flow for a triple (G,I,O) when |I| = |O| = k (corresponding to unitary transformations in the measurement model) and |V(G)| = n can be performed in time O(k^2 n), improving the earlier known bound of O(km) given in [quant-ph/0611284], where m = |E(G)|.