Bass numbers of local cohomology of cover ideals of graphs

We develop splitting techniques to study the Lyubeznik numbers of cover ideals of graphs which allow us to describe them for large families of graphs including forests, cycles, wheels and cactus graphs. More generally, we are able to compute all the Bass numbers and the shape of the injective resolution of local cohomology modules by considering the connected components of all the induced subgraphs. Indeed, our method gives us a very simple criterion for the vanishing of these local cohomology modules in terms of the number of connected components of the induced subgraphs.