  We investigate the conditions required for planet formation via gravitational
instability (GI) and protoplanetary disk (PPD) fragmentation around M-dwarfs.
Using a suite of 64 SPH simulations with $10^6$ particles, the parameter space
of disk mass, temperature, and radius is explored, bracketing reasonable values
based on theory and observation. Our model consists of an equilibrium, gaseous,
and locally isothermal disk orbiting a central star of mass $M_*=M_{sol}/3$.
Disks with a minimum Toomre $Q$ of $Q_{min} \lesssim 0.9$ will fragment and
form gravitationally bound clumps. Some previous literature has found $Q_{min}
< 1.3-1.5$ to be sufficient for fragmentation. Increasing disk height tends to
stabilize disks, and when incorporated into $Q$ as $Q_{eff}\propto
Q(H/R)^\alpha$ for $\alpha=0.18$ is sufficient to predict fragmentation. Some
discrepancies in the literature regarding $Q_{crit}$ may be due to different
methods of generating initial conditions (ICs). A series of 15 simulations
demonstrates that perturbing ICs slightly out of equilibrium can cause disks to
fragment for higher $Q$. Our method for generating ICs is presented in detail.
We argue that GI likely plays a role in PPDs around M-dwarfs and that disk
fragmentation at large radii is a plausible outcome for these disks.
