  In this paper, we consider the impact of Lorentz Invariance Violation (LIV)
on the $\gamma-\gamma$ opacity of the Universe to VHE-gamma rays, compared to
the effect of local under-densities (voids) of the Extragalactic Background
Light, and on the Compton scattering process. Both subluminal and superluminal
modifications of the photon dispersion relation are considered. In the
subluminal case, LIV effects may result in a significant reduction of the
$\gamma-\gamma$ opacity for photons with energies $\gtrsim 10$ TeV. However,
the effect is not expected to be sufficient to explain the apparent spectral
hardening of several observed VHE $\gamma$-ray sources in the energy range from
100 GeV - a few TeV, even when including effects of plausible inhomogeneities
in the cosmic structure. Superluminal modifications of the photon dispersion
relation lead to a further enhancement of the EBL $\gamma\gamma$ opacity. We
consider, for the first time, the influence of LIV on the Compton scattering
process. We find that this effect becomes relevant only for photons at
ultra-high energies, $E \gtrsim 1$ PeV. In the case of a superluminal
modification of the photon dispersion relation, both the kinematic recoil
effect and the Klein-Nishina suppression of the cross section are reduced.
However, we argue that the effect is unlikely to be of astrophysical
significance.
