On graph inverse semigroups

We study the relationship between the graph inverse semigroups of two graphs when there is a directed immersion between the graphs and we provide structural information about graph inverse semigroups of finite graphs that admit a directed cover onto a bouquet of circles. We provide a topological characterization of the universal groups of the local submonoids of a graph inverse semigroup. We also find necessary and sufficient conditions for a homomorphic image of a graph inverse semigroup to be another graph inverse semigroup.