Aspherical connected subgraph arrangements A_G are free, with the possible exception of the K4 graph.
Cuntz, A Greedy Algorithm to Compute Arrangements of Lines in the Projective Plane, Discrete Comput.\ Geom.\ 68 (2022), 107--124
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On connected subgraph arrangements
Aspherical connected subgraph arrangements A_G are free, with the possible exception of the K4 graph.