  Photonic graph states are important for measurement- and fusion-based quantum
computing, quantum networks, and sensing. They can in principle be generated
deterministically by using emitters to create the requisite entanglement.
Finding ways to minimize the number of entangling gates between emitters and
understanding the overall optimization complexity of such protocols is crucial
for practical implementations. Here, we address these issues using graph theory
concepts. We develop optimizers that minimize the number of entangling gates,
reducing them by up to 75$\%$ compared to naive schemes for moderately sized
random graphs. While the complexity of optimizing emitter-emitter CNOT counts
is likely NP-hard, we are able to develop heuristics based on strong
connections between graph transformations and the optimization of stabilizer
circuits. These patterns allow us to process large graphs and still achieve a
reduction of up to $66\%$ in emitter CNOTs, without relying on subtle metrics
such as edge density. We find the optimal emission orderings and circuits to
prepare unencoded and encoded repeater graph states of any size, achieving
global minimization of emitter and CNOT resources despite the average
NP-hardness of both optimization problems. We further study the locally
equivalent orbit of graphs. Although enumerating orbits is $\#$P complete for
arbitrary graphs, we analytically calculate the size of the orbit of repeater
graphs and find a procedure to generate the orbit for any repeater size.
Finally, we inspect the entangling gate cost of preparing any graph from a
given orbit and show that we can achieve the same optimal CNOT count across the
orbit.
