An FPT algorithm for the embeddability of graphs into two-dimensional simplicial complexes

TL;DR

This work resolves the embeddability problem for graphs into 2-dimensional simplicial complexes by proving an FPT algorithm parameterized by the 2-complex size $c$. The authors combine an irrelevant-vertex method with dynamic programming on bounded branchwidth, embedding constraints encoded as sparse bounding graphs, and careful preprocessing to reduce to proper cellular embeddings. The resulting algorithm runs in $O(2^{\text{poly}(c)}\cdot n^2)$ time and can also construct an embedding when one exists; applications include fixed-parameter tractable approaches to the crossing number, planarity number, and embedding-extension problems, with generalization to fixed genus surfaces. This provides a unified, theory-grounded framework that leverages graph minor theory to tackle embeddability in a broad class of host spaces, surpassing prior XP results and linking to major themes in parameterized topology and graph drawing.

Abstract

We consider the embeddability problem of a graph G into a two-dimensional simplicial complex C: Given G and C, decide whether G admits a topological embedding into C. The problem is NP-hard, even in the restricted case where C is homeomorphic to a surface. We prove that the problem is fixed-parameter tractable in the size of the two-dimensional complex, by providing an O(2^{poly(c)}.n^2)-time algorithm. If G embeds into C, we can compute a representation of an embedding in the same amount of time. Moreover, we show that several known problems reduce to this one, such as the crossing number and the planarity number problems, and, under some conditions, the embedding extension problem. Our approach is to reduce to the case where G has bounded branchwidth via an irrelevant vertex method, and to apply dynamic programming. We do not rely on any component of the existing linear-time algorithms for embedding graphs on a fixed surface, but only on algorithms from graph minor theory. However, by combining our results with a linear-time algorithm for embedding graphs on surfaces and with a very recent result for the irrelevant vertex method, we can decide whether G embeds into C in f(c).O(n) time, for some function f.

Figures (10)

• Figure 3.1: On the left: A 2-complex with 5 singular vertices, numbered from 1 to 5, and 2 isolated edges (one between 3 and 4 and one between 1 and 2) where, at singular vertices, the cones are in green and the corners in yellow. On the right: the corresponding detached surface.
• Figure 3.2: On the left: The same 2-complex as Figure \ref{['F:Detached']}. On the right: the corresponding surface constructed in Lemma \ref{['L:genus-oversurface']}.
• Figure 4.1: Construction of the partitioning graph $\Pi=\Pi(\Gamma,E_1,E_2)$, for three choices of the partition $(E_1,E_2)$ of the same embedding $\Gamma$. Only a part of the 2-complex $\mathscr{C}$ is shown, with a boundary at the upper part, and without singular vertex. Left: The graph embeddings $\Gamma$ (in thick lines) and $\Pi$ (in thin lines). Right: The sole graph $\Pi$, together with the labelling of its faces.
• Figure 4.2: The partitioning graph $\Pi=\Pi(\Gamma,E_1,E_2,E_3)$. Left: The graph embeddings $\Gamma$ (in thick lines) and $\Pi$ (in thin lines). Right: The sole graph $\Pi$, together with the labelling of its faces.
• Figure 4.3: A simplification. Left: A vertex of an embedding $\Gamma$ of $G$, on the boundary of $\mathscr{C}$, incident with eight intervals. The exterior intervals are drawn in thick lines. Right: The result of the simplification that exchanges the intervals marked with an arrow. The number of intervals strictly decreases.

Theorems & Definitions (53)

• Theorem 1.1
• Theorem 1.2: algorithm for bounded branchwidth
• Theorem 1.3: algorithm to reduce branchwidth
• Proposition 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4