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

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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 $\lvert I \rvert = \lvert O \rvert = k$ (corresponding to unitary transformations in the measurement model) and $\lvert V(G) \rvert = n$ can be performed in time $O(k^2 n)$, improving the earlier known bound of $O(km)$ given in \cite{B06a}, where $m = \lvert E(G) \rvert$.