A \(\beta \)-perfect graph is a simple graph \(G\) such that \(\chi (G')=\beta (G')\) for every induced subgraph \(G'\) of \(G\), where \(\chi (G')\) is the chromatic number of \(G'\), and \(\beta (G')\) is defined as the maximum over all induced subgraphs \(H\) of \(G'\) of the minimum vertex degree in \(H\) plus 1 (i.e., \(\delta (H)+1\)). The vertices of a \(\beta \)-perfect graph \(G\) can be colored with \(\chi (G)\) colors in polynomial time (greedily). The main purpose of this paper is to give necessary and suficient conditions, in terms of forbidden induced subgraphs, for a graph to be \(\beta \)-perfect. New sufficient conditions and improvements to sufficient conditions previously proposed are given. A necessary condition which generalizes the fact that no \(\beta \)-perfect graph contains an even hole, in terms of forbidden induced subgraphs, is also mentioned.