The Giants gave up more yards in 2012 than in any season in franchise history. Their run defense was beyond shabby, allowing 129.1 yards per game — 25th in the league. It is no secret the major emphasis this off-season is to upgrade the defense, and the Giants took a step in that direction yesterday by agreeing to terms with defensive tackle Cullen Jenkins.

Given his NFL pedigree, Jenkins will move in as a starter alongside Linval Joseph. Jenkins, 32, was cut by the Eagles on Feb. 25, immediately making him an unrestricted free agent, which is why the Giants could bring him in for a visit — which they did last week.

A three-year deal was finalized and Jenkins’ contract is for $8 million, with $3 million in guaranteed money, including a $2 million signing bonus. Consider Jenkins a cheaper alternative to Chris Canty, who was scheduled to make $6 million in salary before he was released.

Jenkins is expected to sign the deal today.

Six teams expressed interest in the 6-foot-2, 305-pound Jenkins, who last week made visits to Seattle, San Francisco and Oakland. But the Giants made the most sense, financially and in terms of having an open slot for him on the field.

There’s also a previous connection with Jenkins and the Giants as their defensive-line coach, Robert Nunn, was Jenkins’ position coach with the Packers from 2005 to 2008. Jenkins after seven seasons with the Packers — following a career-high seven sacks in 2010 — signed a five-year, $25 million deal with the Eagles as part of the supposed “Dream Team’’ they assembled for the 2011 season. Jenkins started every game the past two years and had 9.5 sacks. Jenkins is the younger brother of Kris Jenkins, a former Jets defensive lineman.