\stackrel{*}{\rightarrow} is the transitive closure of \rightarrow \cup =, where = is the identity relation, i.e. \stackrel{*}{\rightarrow} is the smallest preorder (reflexive and transitive relation) containing \rightarrow.
\stackrel{*}{\leftrightarrow} is the transitive closure of \leftrightarrow \cup =, that is \stackrel{*}{\leftrightarrow} is the smallest equivalence relation containing \rightarrow.