  We call a first-order formula one-dimensional if its every maximal block of
existential (universal) quantifiers leaves at most one variable free. We
consider the one-dimensional restrictions of the guarded fragment, GF, and the
tri-guarded fragment, TGF, the latter being a recent extension of GF in which
quantification for subformulas with at most two free variables need not be
guarded, and which thus may be seen as a unification of GF and the two-variable
fragment, FO2. We denote the resulting formalisms, resp., GF1, and TGF1. We
show that GF1 has an exponential model property and NExpTime-complete
satisfiability problem (that is, it is easier than full GF). For TGF1 we show
that it is decidable, has the finite model property, and its satisfiability
problem is TwoExpTime-complete (NExpTime-complete in the absence of equality).
All the above-mentioned results are obtained for signatures with no constants.
We finally discuss the impact of their addition, observing that constants do
not spoil the decidability but increase the complexity of the satisfiability
problem.
