Aspherical connected subgraph arrangements A_G are free, with the possible exception of the K4 graph.
Randell, Lattice-isotopic Arrangements Are Topologically Isomorphic, Proc.\ Amer.\ Math.\ Soc.\ 107 (1989), no
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On connected subgraph arrangements
Aspherical connected subgraph arrangements A_G are free, with the possible exception of the K4 graph.