Your particular question (do halves or thirds) probably needs usability testing.
But if you are talking about mobile-mobile, like tiny-butt phones mobile, another possibility you should be looking at is simply
fewer menu items in total
This means restructuring who link to what and how users would reach a given area.
If you have dedicated mobile content though, you restructure your navigation if it's already quite big. Three, Four, Five items, not 10. There are some usability studies done by Nielsen Group that showed some companies, in their dedicated mobile sites or apps, removing many menu items and renaming some others so they could include more types of content.
I believe one example in the usability study was a responsive site called propublica.org, where at the time of the study, several menu items were simply removed on mobile.
On mobile, because screens are smaller, users do expect to click a few more times to get to stuff, and that's okay. A click on a More button my bring up a dedicated page showing only more menu items, and from there the user has to click again to get to the page they could have gotten to with one click on teh Desktop version.