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

A Weyl arrangement is the hyperplane arrangement defined by a root system. Arnold and Saito proved that every Weyl arrangement is free. The Weyl subarrangements of type $A_{\ell}$ are represented by simple graphs. Stanley gave a characterization of freeness for this type of arrangements in terms of thier graph. In addition, The Weyl subarrangements of type $B_{\ell}$ 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 $ A_{\ell-1} $ and type $ B_{\ell} $. In this paper, we give a characterization of the freeness and supersolvability of the Weyl subarrangements of type $B_{\ell}$ under certain assumption.