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De Simone showed that prime bull-and chair-free graphs containing a co-diamond are either bipartite or an induced cycle of odd length at least ÿve. Based on this result, we give a complete structural characterization of prime (bull,chair)-free graphs having stability number at least four as well as of (bull,chair,co-chair)-free graphs. This implies constant-bounded clique width for these graph classes which leads to linear time algorithms for some algorithmic problems. Moreover, we obtain adoi:10.1016/s0166-218x(02)00415-8 fatcat:vjokgbfafzbltiuxl66ercn5zu