  A \emph{fair competition}, based on the concept of envy-freeness, is a
non-eliminating competition where each contestant (team or individual player)
may not play against all other contestants, but the total difficulty for each
contestant is the same: the sum of the initial rankings of the opponents for
each contestant is the same. Similar to other non-eliminating competitions like
the Round-robin competition or the Swiss-system competition, the winner of the
fair competition is the contestant who wins the most games. The {\sc Fair
Non-Eliminating Tournament} ({\sc Fair-NET}) problem can be used to schedule
fair competitions whose infrastructure is known. In the {\sc Fair-NET} problem,
we are given an infrastructure of a tournament represented by a graph $G$ and
the initial rankings of the contestants represented by a multiset of integers
$S$. The objective is to decide whether $G$ is \emph{$S$-fair}, i.e., there
exists an assignment of the contestants to the vertices of $G$ such that the
sum of the rankings of the neighbors of each contestant in $G$ is the same
constant $k\in\mathbb{N}$. We initiate a study of the classical and
parameterized complexity of {\sc Fair-NET} with respect to several central
structural parameters motivated by real world scenarios, thereby presenting a
comprehensive picture of it.
