We study the Maximum Bipartite Subgraph () problem, which is defined as follows. Given a set of geometric objects in the plane, we want to compute a maximum-size subset such that the intersection graph of the objects in is bipartite. We first give an -time algorithm that computes an almost optimal solution for the problem on circular-arc graphs. We show that the problem is -hard on geometric graphs for which the maximum independent set is -hard (hence, it is -hard even on unit squares and unit disks). On the other hand, we give a for the problem on unit squares and unit disks. Moreover, we show fast approximation algorithms with small-constant factors for the problem on unit squares, unit disks, and unit-height axis parallel rectangles. Additionally, we prove that the Maximum Triangle-free Subgraph () problem is NP-hard for axis-parallel rectangles. Here the objective is the same as that of the except the intersection graph induced by the set needs to be triangle-free only (instead of being bipartite).