We face the problem of devising optimal unification operators for
sharing and linearity analysis of logic programs by abstract
interpretation. We propose a new (infinite) domain ShLinOmega which
can be thought of as a general framework from which other domains can
be easily derived by abstraction. The advantage is that ShLinOmega is
endowed with very elegant and optimal abstract operators for
unification and matching, based on a new concept of sharing graph
which plays the same role of alternating paths for pair sharing
analysis. We also provide an alternative, purely algebraic description
of sharing graphs.  Starting from the results for ShLinOmega, we
derive optimal abstract operators for two well-known domains which
combine sharing and linearity: ShLin by Andy King and the classic
Sharing X Lin.