The topology of the configuration spaces and the independence complexes for digraphs

We study the topology of the (path) configuration spaces and the (path) independence complexes for digraphs as well as their underlying graphs. We prove some canonical embeddings from the (path) independence complexes of the underlying graphs to the (path) independence complexes of the digraphs as well as some canonical embeddings between the (path) independence complexes induced by strong totally geodesic immersions of (di)graphs. We apply the path complexes and the path homology to the path independence complexes of (di)graphs. As by-products, we give some consequences about the Shannon capacities.