 This result implies a direct analysis of the 4/9 - approximation algorithm for maximum planar subgraph problem without using the above min-max formula.An important conjecture related to the triangular cactus is the Rosa's Conjecture, named after Alexander Rosa, which says that all triangular cacti are graceful or nearly-graceful.More precisely 

"All triangular cacti with t ≡ 0, 1 mod 4 blocks are graceful, and those with t ≡ 2, 3 mod 4 are near graceful."