The subgraph homeomorphism problem is to decide whether there is an injective mapping of the vertices of a pattern graph into vertices of a host graph so that the edges of the pattern graph can be mapped into (internally) vertex-disjoint paths in the host graph. The restriction of subgraph homeomorphism where an injective mapping of the vertices of the pattern graph into vertices of the host graph is already given is termed fixed-vertex subgraph homeomorphism. We show that fixed-vertex subgraph homeomorphism for a pattern graph on p vertices and a host graph on n vertices can be solved in time O(2n - pnO(1)) or in time O(3n - pn6) and polynomial space. In effect, we obtain new non-trivial upper time-bounds on the exact complexity of the problem of finding k vertex-disjoint paths and general subgraph homeomorphism.
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