A decision problem has good characterization if it is in $NP \cap coNP$. Many natural graph problems have good characterizations. For instance, Kuratuwski's Theorem gives good characterization of planar graphs. Konig's Theorem gives good characterization of bipartite graphs. Tutte's Theorem gives good characterization of graphs that have perfect matching. Euler's Theorem gives good characterization of Eulerian graphs. All these recognition problems have polynomial-time algorithms.
Is there a natural graph problem that has good characterization but not known to be in $P$? A pointer to a survey of such problems would be appreciated.