1-inkodt alternationg pushdown automaton is a slightly mo-dified alternationg pushdown automaton iwht the additional power of marking at most 1 tape-oell on the input (with an inkodt) once. This paper investigates the closute property of sublgarithmic space-bounded 1-inkodt alternationg pushdown automata with only existential (universal) states, and shows, for example, that for any function L (n) such that L (n)≥log logn and L (n)=o (logn), the class of sets accepted by weakly (strongly) L (n) space-bounded 1-inkodot two-way alternating pushdown automata with only existential (universal) states not closed under concatnation with regular sets, length-preserving homomrphism, and kleene closure.