The complexity of the maximum common connected subgraph problem in partial -trees is still not fully understood. Polynomial-time solutions are known for degree-bounded outerplanar graphs, a subclass of the partial -trees. On the other hand, the problem is known to be NP-hard in vertex-labeled partial -trees of bounded degree. We consider series–parallel graphs, i.e., partial -trees. We show that the problem remains NP-hard in biconnected series–parallel graphs with all but one vertex of degree or less. A positive complexity result is presented for a related problem of high practical relevance which asks for a maximum common connected subgraph that preserves blocks and bridges of the input graphs. We present a polynomial time algorithm for this problem in series–parallel graphs, which utilizes a combination of BC- and SP-tree data structures to decompose both graphs.

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