  We study N=1 SUSY theories in four dimensions with multiple discrete vacua,
which admit solitonic solutions describing segments of domain walls meeting at
one-dimensional junctions. We show that there exist solutions preserving one
quarter of the underlying supersymmetry -- a single Hermitian supercharge. We
derive a BPS bound for the masses of these solutions and construct a solution
explicitly in a special case. The relevance to the confining phase of N=1 SUSY
Yang-Mills and the M-theory/SYM relationship is discussed.
