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The largely coastal district has voted overwhelmingly Republican in recent elections. Representative Nancy Mace ran unsuccessfully for governor instead of seeking re-election to the seat.
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Mr. Graham, a four-term incumbent, is the favorite to win the general election in a solidly Republican state.
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Lt. Gov. Pamela Evette had gotten an endorsement from President Trump, but it was not enough to help her clinch a victory outright. She now faces Alan Wilson, the state attorney general, in a runoff on June 23.
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