Signed graphs and the freeness of the Weyl subarrangements of type [Formula presented]

A Weyl arrangement is the hyperplane arrangement defined by a root system. Saito proved that every Weyl arrangement is free. The Weyl subarrangements of type [Formula presented] are represented by simple graphs. Stanley gave a characterization of freeness for this type of arrangements in terms of their graph. In addition, the Weyl subarrangements of type [Formula presented] can be represented by signed graphs. A characterization of freeness for them is not known. However, characterizations of freeness for a few restricted classes are known. For instance, Edelman and Reiner characterized the freeness of the arrangements between type [Formula presented] and type [Formula presented]. In this paper, we give a characterization of the freeness and supersolvability of the Weyl subarrangements of type [Formula presented] under certain assumption.