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A Weyl arrangement is the hyperplane arrangement defined by a root system.Arnold and Saito proved that every Weyl arrangement is free. The Weylsubarrangements of type $A_{\ell}$ are represented by simple graphs. Stanleygave a characterization of freeness for this type of arrangements in terms ofthier graph. In addition, The Weyl subarrangements of type $B_{\ell}$ can berepresented by signed graphs. A characterization of freeness for them is notknown. However, characterizations of freeness for a few restricted classes areknown. For instance, Edelman and Reiner characterized the freeness of thearrangements between type $ A_{\ell-1} $ and type $ B_{\ell} $. In this paper,we give a characterization of the freeness and supersolvability of the Weylsubarrangements of type $B_{\ell}$ under certain assumption.

Signed graphs and the freeness of the Weyl subarrangements of type $B_{\ell}$