Network camera crews prepare for decisions being issued from the Supreme Court on June 6. (J. Scott Applewhite/AP)

The Supreme Court on Monday said it would review two cases brought by African Americans on death row in Texas, the state that most often carries out what is an increasingly rare punishment in the United States.

One challenge involves Texas’s use of what lawyers say is an outdated method of determining whether inmates’ intellectual disabilities are so great that they should not be executed. The other alleges that racial discrimination infected an inmate’s sentencing.

Last term, Justices Stephen G. Breyer and Ruth Bader Ginsburg urged the court to accept a case that would allow the justices to fully examine whether capital punishment can be carried out in a way that satisfies the Constitution. So far, the court has not accepted such a case.

But Monday’s actions and recent decisions by the court are a reminder of how much time the court spends policing the imposition of the death penalty and the inclination of the justices to put limits on its use.

The court last term upheld the use of a much criticized drug used in lethal injections. But it has also thrown out the death sentence imposed on a black Georgia man by an all-white jury, tossed out the capital-punishment sentencing scheme in Florida, raised questions about the method used in Alabama and said states must be more flexible in determining intellectual disability.

Both of the cases accepted Monday come from Houston, which is in Harris County. About half of the African American prisoners on Texas’s death row are from there, according to lawyers in the case. They add that since December 2004, all of the new death sentences in Harris County have been imposed on men of color — three Hispanic men and 13 black men.

Texas has carried out six executions this year, representing nearly half of the 14 executions nationwide.

One of the cases is brought by Duane Buck, who says his trial attorney was inadequate because he presented testimony at sentencing by an expert who said black people were more likely to be violent.

“By any measure, Duane Buck’s death sentence is extraordinary,” his new attorneys said in a brief to the Supreme Court. “At sentencing, his trial attorney presented ‘bizarre and objectionable’ testimony from a ‘defense expert’ that Mr. Buck was more likely to be dangerous in the future because he is black.”

Buck was convicted of murder in the killing of his ex-girlfriend and a man at the woman’s Houston apartment in July 1995; her children were present.

During the sentencing phase of his trial, Buck’s attorney called psychologist Walter Quijano to testify about the convicted man’s “future dangerousness,” something Texas requires in order to impose death.

Buck’s attorney put Quijano on the stand even though he knew that Quijano believed “that race was among the ‘statistical factors in deciding whether a person will or will not constitute a continued danger,’ ” said Buck’s brief to the Supreme Court, filed by the NAACP Legal Defense and Educational Fund and the Texas Defender Service.

Then, on cross-examination, “the trial prosecutor exploited and compounded defense counsel’s error by asking Dr. Quijano to reiterate his false and discriminatory ‘expert’ opinion that Mr. Buck’s race increased his likelihood of future dangerousness,” the brief said.

Then-Texas Attorney General John Cornyn, now a Republican U.S. senator, conducted a study and said six death-penalty cases, including Buck’s, needed to be reopened because of testimony by Quijano.

But the U.S. Court of Appeals for the 5th Circuit denied Buck’s attempt to overturn his death sentence. Texas says future litigation of the long-running case is barred for procedural reasons.

Bobby J. Moore’s case is even older. He was convicted of murder for a killing during a grocery-store robbery in Houston in 1980. His case raises the question of whether modern standards should be used in determining whether he is intellectually disabled and thus ineligible for the death penalty.

Moore’s attorneys say the Texas Court of Criminal Appeals ignored the Supreme Court’s recent rulings about not using a strict IQ-score threshold in determining whether an inmate is mentally incompetent.

“Texas’s highest criminal court has decided that Texas courts must continue to apply a 1992 standard of intellectual disability unless and until the state’s legislature sees fit to enact” new regulations, said the brief filed by Washington lawyer Clifford M. Sloan. That approach “defies both the Constitution and common sense,” it said.

When the court first announced Monday morning it was accepting Moore’s case, it indicated it would also consider whether Moore’s 35 years on death row and 15 years in solitary confinement violated the Constitution’s protection against cruel and unusual punishment.

But hours later it clarified that the justices were considering only the intellectual-disability question.

The cases are Buck v. Stephens and Moore v. Texas.