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Given graphs G and H with V(G) containing V(H), suppose that we have au,v-path P_{uv} in G for each edge uv in H. There are obvious additionalconditions that ensure that G contains H as a rooted subgraph, subdivision, orimmersion; we seek conditions that ensure that G contains H as a rooted minoror minor. This naturally leads to studying sets of paths that form anH-immersion, with the additional property that paths that contain the samevertex must have a common endpoint. We say that $H$ is contractible if,whenever G contains such an H-immersion, G must also contain a rooted H-minor.We show, for example, that forests, cycles, K_4, and K_{1,1,3} arecontractible, but that graphs that are not 6-colorable and graphs that containcertain subdivisions of K_{2,3} are not contractible.

Finding minors in graphs with a given path structure