The Lone Star State's mammoth number of executions has been under national scrutiny lately, because of the Graham case and the GOP presidential bid of Texas Governor George W. Bush.
On CBS News' Face The Nation on Sunday, the Rev. Jesse Jackson, who was present for Graham's execution last week, said it's time for a national moratorium on the death penalty, given what he called " the flawed system that is now embarrassing us as a nation."
But Rep. Asa Hutchinson (R-Ark.) said a moratorium would be "the wrong direction" to take, because that step would be too sweeping.
"Whenever the evidence is absolutely clear and it has been reviewed by the courts, I don't think that we ought to have a moratorium," he told Face The Nation. "I think what we have to do is to make sure that the system works in each individual case."
So instead, Hutchinson said, states with the death penalty should decide whether to impose a moratorium, such as Illinois has done. As for the federal government, its role should be to provide states with more money for DNA testing and for better legal representation.
Jackson pointed out that the court-appointed lawyer that Graham got shows why nothing less than a moratorium now is a must.
"In this case, a politically elected judge appointed a lawyer - an unqualified lawyer who slept through the case, and there was never cross-examination of other witnesses. He was killed on the strength of one witness "
Hutchinson countered that the Graham case was thoroughly reviewed by the courts over the last 19 years.
"Over 30 judges have looked at this. They looked at ineffective assistant of counsel and (T)he federal court system said: the system has worked, the jury made a right decision, he had effective representation of counsel."
And what about televising executions, as Jackson has proposed?
"I don't think that bringing this into the living rooms of our children, our families, is the right thing to do," said Hutchinson. "If you look at the vicious murders that are accomplished in our society, I don't think we have to televise and see every detail of that to know the terrible violence that is being happening."
For Jackson, the idea is about walking the talk about capital punishment.
"If you want it to be a deterrent, let the world see the whole process play out. You know, Pontius Pilate, he washed his hands; he wouldn't go to the scene. I say, don't wash your hands; be courageous. If you believe Texas is a safer state today, you should see it for yourself."