# Embedding graphs into two-dimensional simplicial complexes

## TL;DR

This paper studies the problem of embedding graphs into two-dimensional simplicial complexes, providing a polynomial-time solution for fixed complexes and connecting it to existing surface embedding algorithms.

## Contribution

It introduces a polynomial-time algorithm for embedding graphs into fixed 2D complexes by reducing the problem to an embedding extension problem on surfaces.

## Key findings

- NP-complete when the complex is part of the input
- Polynomial-time algorithm for fixed complexes
- Reduces to an embedding extension problem on surfaces

## Abstract

We consider the problem of deciding whether an input graph G admits a topological embedding into a two-dimensional simplicial complex C. This problem includes, among others, the embeddability problem of a graph on a surface and the topological crossing number of a graph, but is more general. The problem is NP-complete when C is part of the input, and we give a polynomial-time algorithm if the complex C is fixed. Our strategy is to reduce the problem to an embedding extension problem on a surface, which has the following form: Given a subgraph H' of a graph G', and an embedding of H' on a surface S, can that embedding be extended to an embedding of G' on S? Such problems can be solved, in turn, using a key component in Mohar's algorithm to decide the embeddability of a graph on a fixed surface (STOC 1996, SIAM J. Discr. Math. 1999).