  Graph states are key resources for measurement-based quantum computing which
is particularly promising for photonic systems. Fusions are probabilistic Bell
state measurements, measuring pairs of parity operators of two qubits. Fusions
can be used to connect/entangle different graph states making them a powerful
resource for measurement-based and the related fusion-based quantum computing.
There are several different graph structures and types of Bell state
measurements, yet the associated graph transformations have only been analyzed
for a few specific cases. Here, we provide a full set of such graph
transformation rules and we give an intuitive visualization based on Venn
diagrams of local neighborhoods of graph nodes. We derive these graph
transformations for all fusion types showing that there are five different
types of fusion success cases. Finally, we give application examples of the
derived graph transformation rules and show that they can be used for
constructing graph codes or simulating fusion networks.
