The San Angelo Standard-Times reported 42 potential jurors from an initial group of about 140 were sent home in the morning. A similar-sized group went through the preliminary screening process during the afternoon.
On Tuesday, more than 200 potential jurors will report back for the next stage in the selection process in which prosecution and defense attorneys will vet them based on questionnaires the group filled out Monday.
The work of paring down the jury pool began after the presiding judge refused to delay Jeffs' trial.
Judge Barbara Walther rejected a motion for a 180-day continuance by newly hired defense attorney Deric Walpole, the newspaper said.
Texas Attorney General Greg Abbott attended the trial opening.
Jeffs sat at the defense table in a suit and tie instead of his usual prison jumpsuit.
Jeffs, leader of the Fundamentalist Church of Jesus Christ of Latter Day Saints, a Mormon splinter sect, is charged with sexual assault of a child in 2005 and aggravated sexual assault of a child in 2006. He could face 119 years in prison.
The aggravated sexual assault count asserts that he married a 12-year-old girl and fathered a child with her. Under Texas law, a girl that age doesn't have the legal right to give consent for sex.
Jeffs' West Texas compound was raided by authorities in 2008.
The Salt Lake Tribune said Jeffs, 55, has received thousands of letters from followers since his incarceration.
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