  Cluster states and graph states in general offer a useful model of the
stabilizer formalism and a path toward the development of measurement-based
quantum computation. Their defining structure -- the stabilizer group --
encodes all possible correlations which can be observed during measurement.
Those outcomes which are compatible with the stabilizer structure make error
correction possible. Here, we leverage both properties to design feasible
families of states that can be used as robust building blocks of quantum
computation. This procedure reduces the effect of experimentally relevant noise
models on the extraction of smaller entangled states from the larger noisy
graph state. In particular, we study the extraction of Bell pairs from linearly
extended graph states -- this has the immediate consequence for state
teleportation across the graph. We show that robust entanglement can be
extracted by proper design of the linear graph with only a minimal overhead of
the physical qubits. This scenario is relevant to systems in which the
entanglement can be created between neighboring sites. The results shown in
this work may provide a mathematical framework for noise reduction in
measurement-based quantum computation. With proper connectivity structures, the
effect of noise can be minimized for a large class of realistic noise
processes.
