  To assess how external factors such as local interactions and fresh gas
accretion influence the global ISM of galaxies, we analyze the relationship
between recent enhancements of central star formation and total
molecular-to-atomic (H2/HI) gas ratios, using a broad sample of field galaxies
spanning early-to-late type morphologies, stellar masses of 10^(7.2-11.2) Msun,
and diverse stages of evolution. We find that galaxies occupy several loci in a
"fueling diagram" that plots H2/HI vs. mass-corrected blue-centeredness, a
metric tracing the degree to which galaxies have bluer centers than the average
galaxy at their stellar mass. Spiral galaxies show a positive correlation
between H2/HI and mass-corrected blue-centeredness. When combined with previous
results linking mass-corrected blue-centeredness to external perturbations,
this correlation suggests a link between local galaxy interactions and
molecular gas inflow/replenishment. Intriguingly, E/S0 galaxies show a more
complex picture: some follow the same correlation, some are quenched, and a
distinct population of blue-sequence E/S0 galaxies (with masses below key
transitions in gas richness) defines a separate loop in the fueling diagram.
This population appears to be composed of low-mass merger remnants currently in
late- or post-starburst states, in which the burst first consumes the H2 while
the galaxy center keeps getting bluer, then exhausts the H2, at which point the
burst population reddens as it ages. Multiple lines of evidence suggest
connected evolutionary sequences in the fueling diagram. In particular,
tracking total gas-to-stellar mass ratios within the diagram provides evidence
of fresh gas accretion onto low-mass E/S0s emerging from central starbursts.
Drawing on a comprehensive literature search, we suggest that virtually all
galaxies follow the same evolutionary patterns found in our broad sample.
