\textit{C. Yuting} and \textit{M. Kano} [J. Graph Theory 12, No. 3, 327-333 (1988; Zbl 0661.05049)] gave a necessary and sufficient condition for \(G\) to have a \([1,f]\)-odd factor (where given \(f:v(G) \to \{1,3,5, \dots\}\), a \([1,f]\)-odd factor is a subgraph \(H\) of \(G\) so that \(\deg_ H (x) \in \{1,2, \dots, f(x)\}\) for all \(x \in V(G))\). As is typical for this type of result an inequality must be checked for each subset of \(V(G)\). In the case when \(G\) is a tree they show that only \(| V(G) |\) inequalities need to be checked. The authors of the paper under review are interested in discovering for which graphs fewer inequalities can be considered. They do this by studying barrier sets. In particular, they show: (1) If \(G\) has even order and is \(n\)-connected it has a \([1,f]\)-odd factor if no vertex of \(G\) is the centre of an induced star \(K_{1,nf(v)+1}\); (2) If \(G\) is a connected block graph of even order only \(| V(G) |\) inequalities need be checked; and (3) If \(G\) is a unicyclic graph of even order at most \({| V (G) | \choose 2}\) inequalities need be checked. Results for cacti are also obtained.