On Maximum Common Subgraph Problems in Series-Parallel Graphs

TL;DR

This paper investigates the computational complexity of the maximum common connected subgraph problem in series-parallel graphs, showing NP-hardness in general but providing a polynomial-time solution for a related problem that preserves graph blocks and bridges.

Contribution

It proves NP-hardness of the problem in biconnected series-parallel graphs and offers a polynomial-time algorithm for a practical variant that maintains blocks and bridges.

Findings

NP-hardness in biconnected series-parallel graphs with degree constraints

Polynomial-time algorithm for block-bridge preserving subgraph problem in series-parallel graphs

Utilization of BC- and SP-tree data structures for decomposition

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 $2$-trees. On the other hand, the problem is known to be ${\bf NP}$-hard in vertex-labeled partial $11$-trees of bounded degree. We consider series-parallel graphs, i.e., partial $2$-trees. We show that the problem remains ${\bf NP}$-hard in biconnected series-parallel graphs with all but one vertex of degree $3$ 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.