Supersolvability of complementary signed-graphic hyperplane arrangements

We study a class of hyperplane arrangements associated to complementary signed graphs in which the positive part and the negative part are complementary to each other in Kn, the complete graph on n vertices. These arrangements form a subclass of the Dn arrangement but do not contain the An−1 arrangement. The main result says that the arrangement A(G) of a complementary signed graph G is supersolvable if and only if the graph G is switching equivalent to a complementary signed graph with negative part a star. We also prove that if G is not switching equivalent to Ḡ and A(G) is supersolvable, then A(Ḡ) is not supersolvable, where Ḡ is the opposite graph of G. ∗ The first author is supported by NSF of China under grant 10271023, and SRF for ROCS, SEM. † The second author is supported by NSF of China under grant 10071087. 262 GUANGFENG JIANG AND JIANMING YU