Embedding Graphs into Two-Dimensional Simplicial Complexes.

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 into an embedding extension problem on a surface, which has the following form: Given a subgraph Hu0027 of a graph Gu0027, and an embedding of Hu0027 on a surface S, can that embedding be extended to an embedding of Gu0027? Such problems can be solved, in turn, using a key component in Moharu0027s algorithm to decide the embeddability of a graph on a fixed surface (STOC 1996, SIAM J. Discr. Math. 1999).