Extension of the decidability of the marked PCP to instances with unique blocks

In the Post Correspondence Problem (PCP) an instance (h,g)$(h,g)$ consists of two morphisms h$h$ and g$g$, and the problem is to determine whether or not there exists a nonempty word w$w$ such that h(w)=g(w)$h\left(w\right)=g\left(w\right)$. Here we prove that the PCP is decidable for instances with unique blocks using the decidability of the marked PCP. Also, we show that it is decidable whether an instance satisfying the uniqueness condition for continuations has an infinite solution. These results establish a new and larger class of decidable instances of the PCP, including the class of marked instances.