We solve the subgraph isomorphism problem in planar graphs in linear time, for any pattern of constant size. Our results are based on a technique of partitioning the planar graph into pieces of small tree-width, and applying dynamic programming within each piece. The same methods can be used to solve other planar graph problems including connectivity, diameter, girth, induced subgraph isomorphism, and shortest paths.
Eppstein REF gave linear time algorithms for solving the subgraph isomorphism problem for bounded genus graphs.
Subgraph Isomorphism in Planar Graphs and Related Problems
J. Graph Algorithms&Applications 3(3):1-27, 1999
Computer Science Mathematics