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PRINCE SULTAN AIR BASE, Saudi Arabia—Defense Secretary Mark Esper met Tuesday with the king of Saudi Arabia during a tour of the kingdom’s military facilities as the U.S. pursues its biggest Middle East troop buildup of the Trump administration.

Mr. Esper arrived in Saudi Arabia Monday as the U.S. steps up support for Riyadh against a recent series of attacks on Saudi oil industry facilities and tankers blamed on Iran. The trip also comes amid strained U.S.-Saudi relations and growing Russian influence in the region,