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

TL;DR

This paper presents a fixed-parameter tractable algorithm for determining if a graph can be embedded into a two-dimensional simplicial complex, extending the understanding of graph embeddability beyond surfaces.

Contribution

The authors develop an FPT algorithm for graph embeddability into 2D complexes, using irrelevant vertex techniques and dynamic programming, independent of existing surface embedding algorithms.

Findings

Algorithm runs in O(2^{poly(c)}.n^2) time for embedding decision.

Embedding can be computed within the same time complexity.

Reduces several problems like crossing number and planarity number to this embedding problem.

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.