Dennis Tyler never expected to fall in love. But after he took in a former racing greyhound named Clara Voyant in 1991, Tyler soon discovered that he'd found a constant companion in the affectionate light-brown dog, who strode beside him on daily walks and nestled against him while watching TV. "She was a very special dog," recalls the Melbourne, Fla.-based father of two grown children. "She blended right into the family." So when Tyler, 66, learned that many greyhounds, some injured while racing, faced the risk of being euthanized, he couldn't look the other way. "These dogs can be the most wonderful pets,...