But pollster Margaret Kenski of Arizona Opinion cautioned that both races remained up for grabs, noting that none of the candidates "has a magic blend of personality, money, organization and issue command that guarantees election. If we learn nothing else from the next three weeks, it could be the lesson that campaigns do matter."
The poll, taken last week, showed that Graf, a former state lawmaker who captured more than 42 percent of the vote in his unsuccessful primary campaign against retiring Congressman Jim Kolbe two years ago, had the support of 36 percent of 300 Republicans surveyed.
"I think that what Randy has going for him right now is that he has residual name recognition," Kenski says. "The others have to buy it."
Graf's closest rival was state Rep. Steve Huffman, who had the support of 13 percent of Republicans polled. In third place was former Arizona Republican Party Chairman Mike Hellon, with 10 percent.
Auto-shop manager Mike Jenkins was polling at 1 percent, while former Green Beret Frank Antenori, making his first run for public office, was at less than 1 percent.
But with 39 percent of GOP voters undecided, and nearly three weeks left for campaigns to drop both negative and positive ads before the Sept. 12 primary, the numbers could still change dramatically, Kenski says.
"Huffman is engaged in a strategy of negative attacks on Randy," Kenski says. "What does that do? Does it help him? Does it hurt Randy? Or do people throw up their hands with the two of them and vote for Hellon? I don't know."
On the Democratic side, former state lawmaker Giffords had the support of 45 percent of 300 Democrats polled. Coming in second was Patty Weiss, with 27 percent saying they backed the former newscaster. Gulf War vet Jeff Latas had the support of 5 percent, while the remaining candidates--TUSD school board member Alex Rodriguez, retired federal bureaucrat Francine Shacter and attorney Bill Johnson--were each polling at 2 percent or less.
Roughly one in five Democratic voters remained undecided.
Republicans showed more willingness than Democrats to cross party lines in the general election if their preferred candidate loses the primary. More than 21 percent of GOP voters said they'd consider voting for the Democratic candidate in November, compared to 16 percent of Democrats. Another 21 percent of Republicans were undecided on the question, compared to 16 percent of Democrats.
"That's unusual," Kenski says. "Historically, Republicans have been a bit more loyal to their party than Democrats have been to theirs."
In particular, while 64 percent of Graf supporters said they'd stick with the eventual Republican nominee if Graf did not win the primary, only 57 percent of Republicans who were supporting another candidate said they would definitely support the GOP nominee if their preferred candidate didn't prevail.
The poll also showed that primary voters have dramatically different concerns. More than 60 percent of Republicans identified border security as their top concern, while less than 16 percent of Democrats did. Nearly 40 percent of Democrats identified the war in Iraq as their top priority, compared to 16 percent of Republicans.
Kenski suggests that split will present challenges for the eventual winners of both primaries.
"Clearly, whoever the Republican nominee is, he will have to reassure the base on the border, but define himself on the other major issues," Kenski says. "Similarly, the Democrat will have to address the border issue forthrightly if he or she is to gain crossover Republicans and independents."
In an open-ended question asking why voters were supporting their preferred candidate, 16 percent of Graf supporters said they were familiar with him from his 2004 campaign, while 9 percent cited his stand on border issues, and 5 percent said they liked his positions on issues in general.
Roughly 6 percent of Huffman supporters told pollsters they thought he'd done a good job as a state legislator, while 7 percent of Hellon supporters said they liked his advertising message.
Among Democrats, 21 percent of Giffords' supporters said they supported her because of her experience as a state lawmaker, while 5 percent cited her stand on issues in general.
"That shows not only a good use of advertising, but also organizational ability," notes Kenski, who says that legislators are usually not well-known outside of their district (and sometimes not even within their district). "In short, Ms. Giffords has demonstrated abilities in both wholesale and retail politics."
Eight percent of Weiss supporters said they had met her or knew her; 6 percent cited her experience as a newscaster, and 6 percent like her stand on the issues in general.
The poll, which targeted Democrats and Republicans who voted in the 2002 and 2004 primary elections, has a margin of error of plus or minus 4 percent. Conducted Aug. 12 and Aug. 14-17, the survey included 249 Republicans in Pima and Pinal counties and 51 in Cochise and Santa Cruz counties; and 246 Democrats in Pima and Pinal counties and 54 in Cochise and Santa Cruz Counties.