Surface Split Decompositions and Subgraph Isomorphism in Graphs on Surfaces

The Subgraph Isomorphism problem asks, given a host graph G on n vertices and a pattern graph P on k vertices, whether G contains a subgraph isomorphic to P. The restriction of this problem to planar graphs has often been considered. After a sequence of improvements, the current best algorithm for planar graphs is a linear time algorithm by Dorn (STACS ‘10), with complexity $2^{O(k)} O(n)$. We generalize this result, by giving an algorithm of the same complexity for graphs that can be embedded in surfaces of bounded genus. At the same time, we simplify the algorithm and analysis. The key to these improvements is the introduction of surface split decompositions for bounded genus graphs, which generalize sphere cut decompositions for planar graphs. We extend the algorithm for the problem of counting and generating all subgraphs isomorphic to P, even for the case where P is disconnected. This answers an open question by Eppstein (SODA ‘95 / JGAA ‘99).

💡 Research Summary

The paper addresses the Subgraph Isomorphism problem—given a host graph G with n vertices and a pattern graph P with k vertices, decide whether G contains a subgraph isomorphic to P. While the problem is NP‑complete in general, parameterized algorithms that treat k as the parameter have been extensively studied for planar graphs. The best known planar algorithm, due to Dorn (STACS 2010), runs in 2^{O(k)}·O(n) time.

The authors extend this result to graphs that can be embedded on surfaces of bounded genus g. Their main contribution is the introduction of surface split decompositions, a new type of branch decomposition that generalizes the sphere‑cut decompositions used for planar graphs. In a surface split decomposition, each edge of the decomposition tree separates the graph into two subgraphs that lie in two disjoint connected regions of the surface; the only relevant measure is the size of the mid‑set (the vertices shared by the two subgraphs). By ensuring that every mid‑set has size O(k) (and that the constant depends only on g), the authors obtain a decomposition whose width is linear in k.

The paper proves that for any fixed genus g, a surface split decomposition of width O(k) can be constructed in linear time. The construction builds on the Tamaki‑Dorn algorithm for low‑width sphere‑cut decompositions, adapting it to handle the handles of the surface by cutting along a bounded number of nooses and “flattening” the surface locally.

With such a decomposition in hand, the authors apply a dynamic‑programming (DP) scheme that processes the decomposition tree bottom‑up. For each tree node they store a DP table that records, for every possible mapping of pattern vertices to the current mid‑set, whether the corresponding partial embedding can be extended to the subgraph represented by the node. Because the mid‑set size is O(k), the table size is 2^{O(k)}; each DP transition merges two tables in 2^{O(k)} time, leading to an overall running time of 2^{O(k)}·O(n).

A notable extension is the handling of disconnected pattern graphs. The authors decompose P into its connected components, run the DP independently on each component, and then combine the results using inclusion‑exclusion to avoid double‑counting. This yields an algorithm that not only counts all subgraph isomorphisms but also enumerates them in 2^{O(k)}·O(n) + m·k^{O(1)} time, where m is the number of solutions. This resolves an open question posed by Eppstein (SODA 1995 / JGAA 1999) about counting and listing all (possibly disconnected) subgraph isomorphisms in bounded‑genus graphs.

The paper also discusses extensions to non‑orientable surfaces, induced subgraph versions, and the broader impact of surface split decompositions on other FPT problems (e.g., Minimum Independent Set, Maximum Dominating Set) that rely on branch‑decomposition‑based DP. The authors argue that many algorithms previously limited to planar graphs can be lifted to bounded‑genus graphs with only a modest increase in constant factors, thanks to the simpler structure of surface split decompositions compared with earlier “surface‑cut” techniques.

In summary, the authors present a conceptually simpler and theoretically optimal 2^{O(k)}·O(n) algorithm for Subgraph Isomorphism on bounded‑genus graphs, together with counting and enumeration capabilities for disconnected patterns. The work introduces a powerful new decomposition tool that is likely to influence future research on parameterized algorithms for a wide range of problems on sparse graph classes.