Partiality is important in Mathematics and Computer Science. In the context of graphs, categories of partial graphs homomorphisms are usual but we take a different approach: partiality appears in the internal structure of objects, using an extension of Comma Categories. We define the category Grp of partial graphs (arcs with or without source/target nodes) and total morphisms. The generalization of the framework results in categories of internal partial graphs. We show that Grp is bicomplete. Partial graphs can be used to define computational models like automata. A category of partial automata is constructed. Using an extension of Span Composition it is possible to define the computations of automata.