Using a sample of dwarf galaxies observed using the VIMOS IFU on the VLT, we investigate the mass-metallicity relation (MZR) as a function of star formation rate (FMR$_{text{SFR}}$) as well as HI-gas mass (FMR$_{text{HI}}$). We combine our IFU data with a subsample of galaxies from the ALFALFA HI survey crossmatched to the Sloan Digital Sky Survey to study the FMR$_{text{SFR}}$ and FMR$_{text{HI}}$ across the stellar mass range 10$^{6.6}$ to 10$^{8.8}$ M$_odot$, with metallicities as low as 12+log(O/H) = 7.67. We find the 1$sigma$ mean scatter in the MZR to be 0.05 dex. The 1$sigma$ mean scatter in the FMR$_{text{SFR}}$ (0.02 dex) is significantly lower than that of the MZR. The FMR$_{text{SFR}}$ is not consistent between the IFU observed galaxies and the ALFALFA/SDSS galaxies for SFRs lower than 10$^{-2.4}$ M$_odot$ yr$^{-1}$, however this could be the result of limitations of our measurements in that regime. The lowest mean scatter (0.01 dex) is found in the FMR$_{text{HI}}$. We also find that the FMR$_{text{HI}}$ is consistent between the IFU observed dwarf galaxies and the ALFALFA/SDSS crossmatched sample. We introduce the fundamental metallicity luminosity counterpart to the FMR, again characterized in terms of SFR (FML$_{text{SFR}}$) and HI-gas mass (FML$_{text{HI}}$). We find that the FML$_{text{HI}}$ relation is consistent between the IFU observed dwarf galaxy sample and the larger ALFALFA/SDSS sample. However the 1$sigma$ scatter for the FML$_{text{HI}}$ relation is not improved over the FMR$_{text{HI}}$ scenario. This leads us to conclude that the FMR$_{text{HI}}$ is the best candidate for a physically motivated fundamental metallicity relation.
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