  A graph $G$ is well-covered if every minimal vertex cover of $G$ is minimum,
and a graph $G$ is well-dominated if every minimal dominating set of $G$ is
minimum. Studies on well-covered graphs were initiated in [Plummer, JCT 1970],
and well-dominated graphs were first introduced in [Finbow, Hartnell and
Nowakow, AC 1988]. Well-dominated graphs are well-covered, and both classes
have been widely studied in the literature. The recognition of well-covered
graphs was proved coNP-complete by [Chv\'atal and Slater, AODM 1993] and by
[Sankaranarayana and Stewart, Networks 1992], but the complexity of recognizing
well-dominated graphs has been left open since their introduction. We close
this complexity gap by proving that recognizing well-dominated graphs is
coNP-complete. This solves a well-known open question (c.f. [Levit and Tankus,
DM 2017] and [G\"{o}z\"{u}pek, Hujdurovic and Milani\v{c}, DMTCS 2017]), which
was first asked in [Caro, Seb\H{o} and Tarsi, JAlg 1996]. Surprisingly, our
proof is quite simple, although it was a long-standing open problem. Finally,
we show that recognizing well-totally-dominated graphs is coNP-complete,
answering a question of [Bahad\ir, Ekim, and G\"oz\"upek, AMC 2021].
