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(J. Comput. Syst. Sci. <jats:bold>123<\/jats:bold>, 103\u2013121, 2022) show that the stability, vertex stability, and unfrozenness problems with respect to certain graph parameters are complete for <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\varvec{\\Theta _{2}^{\\textrm{P}}}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msubsup>\n                      <mml:mi>\u0398<\/mml:mi>\n                      <mml:mrow>\n                        <mml:mn>2<\/mml:mn>\n                      <\/mml:mrow>\n                      <mml:mtext>P<\/mml:mtext>\n                    <\/mml:msubsup>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, the class of problems solvable in polynomial time by parallel access to an NP oracle. They studied the common graph parameters <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\varvec{\\alpha }$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u03b1<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> (the independence number), <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\varvec{\\beta }$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u03b2<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> (the vertex cover number), <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\varvec{\\omega }$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u03c9<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> (the clique number), and <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\varvec{\\chi }$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u03c7<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> (the chromatic number). We complement their approach by providing polynomial-time algorithms solving these problems for special graph classes, namely for graphs with bounded tree-width or bounded clique-width. 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