  We study Diophantine equations of type $f(x)=g(y)$, where $f$ and $g$ are
lacunary polynomials. According to a well known finiteness criterion, for a
number field $K$ and nonconstant $f, g\in K[x]$, the equation $f(x)=g(y)$ has
infinitely many solutions in $S$-integers $x, y$ only if $f$ and $g$ are
representable as a functional composition of lower degree polynomials in a
certain prescribed way. The behaviour of lacunary polynomials with respect to
functional composition is a topic of independent interest, and has been studied
by several authors. In this paper we utilize known results and develop some new
results on the latter topic.
