Consider a context-free grammar where all rules produce at most one non-terminal (i.e., there is at most one non-terminal on the right-hand side of a rule). What is the class of languages which are accepted by such a grammar?
If you like, we may assume that all rules are one of the following forms, where $V, V'$ denote non-terminals and $a$ denotes a terminal:
(i) $V \to aV'$
(ii) $V \to V' a$
(iii) $V \to \epsilon$
If all rules are of the form (i) or (iii) we get the class of regular languages. On the other hand, we can also recognize more languages, such as the language of palindromes.
Therefore, it is somewhere between regular languages and context-free. Does it perhaps coincide with unambiguous context-free or deterministic context-free?