MIS complexity in cubic triangle-free graphs
The paper &quot;The Vertex-Disjoint Triangles Problem&quot; (http://link.springer.com/chapter/10.1007/10692760_3) by Guruswami et al. has the proof of its NP-completeness as Theorem 3.
Update: As stated in a more recent answer to Complexity of Independent Set on Triangle-Free Planar Cubic Graphs, there is a note &quot;NP-complete problems on a 3-connected cubic planar graph and their applications&quot; by Ryuhei Uehara that contains a proof that MIS is NP-complete for cubic, planar, and 3-connected graphs of girth greater than 3.