We show how to verify four challenging concurrent fine-grained graph-manipulating algorithms, including graph copy, a speculatively parallel Dijkstra, graph marking and spanning tree. We develop a reasoning method for such algorithms that dynamically tracks the contributions and responsibilities of each thread operating on a graph, even in cases of arbitrary recursive thread creation. We demonstrate how to use a logic without abstraction ( Open image in new window ) to carry out abstract reasoning in the style of iCAP, by building the abstraction into the proof structure rather than incorporating it into the semantic model of the logic.