  A graph is well-covered if all its maximal independent sets have the same
cardinality. This well studied concept was introduced by Plummer in 1970 and
naturally generalizes to the weighted case. Given a graph $G$, a real-valued
vertex weight function $w$ is said to be a well-covered weighting of $G$ if all
its maximal independent sets are of the same weight. The set of all
well-covered weightings of a graph $G$ forms a vector space over the field of
real numbers, called the well-covered vector space of $G$. Since the problem of
recognizing well-covered graphs is $\mathsf{co}$-$\mathsf{NP}$-complete, the
problem of computing the well-covered vector space of a given graph is
$\mathsf{co}$-$\mathsf{NP}$-hard. Levit and Tankus showed in 2015 that the
problem admits a polynomial-time algorithm in the class of claw-free graph. In
this paper, we give two general reductions for the problem, one based on
anti-neighborhoods and one based on modular decomposition, combined with
Gaussian elimination. Building on these results, we develop a polynomial-time
algorithm for computing the well-covered vector space of a given fork-free
graph, generalizing the result of Levit and Tankus. Our approach implies that
well-covered fork-free graphs can be recognized in polynomial time and also
generalizes some known results on cographs.
