  We study the Balanced Connected Subgraph(shortly, BCS) problem on geometric
intersection graphs such as interval, circular-arc, permutation, unit-disk,
outer-string graphs, etc. Given a vertex-colored graph $G=(V,E)$, where each
vertex in $V$ is colored with either ``red'' or ``blue'', the BCS problem seeks
a maximum cardinality induced connected subgraph $H$ of $G$ such that $H$ is
color-balanced, i.e., $H$ contains an equal number of red and blue vertices. We
study the computational complexity landscape of the BCS problem while
considering geometric intersection graphs. On one hand, we prove that the BCS
problem is NP-hard on the unit disk, outer-string, complete grid, and unit
square graphs. On the other hand, we design polynomial-time algorithms for the
BCS problem on interval, circular-arc and permutation graphs. In particular, we
give algorithm for the Steiner Tree problem on both the interval graphs and
circular arc graphs, that is used as a subroutine for solving BCS problem on
same graph classes. Finally, we present a FPT algorithm for the BCS problem on
general graphs.
