# On Maximum Common Subgraph Problems in Series-Parallel Graphs

## Abstract

The complexity of the maximum common connected subgraph problem in partial k-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.

## BibTeX

```
@article{Kurpicz2018EurJComb,
author = {Nils M. Kriege and Florian Kurpicz and Petra Mutzel},
title = {On Maximum Common Subgraph Problems in Series-Parallel Graphs},
booktitle = {Eur. J. Comb.},
pages = {79–95},
publisher = {SIAM},
doi = {10.1016/j.ejc.2017.07.012},
year = {2018}
}
```