Embedding graphs into two-dimensional simplicial complexes
نویسندگان
چکیده
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). 2012 ACM Subject Classification Theory of computation → Randomness, geometry and discrete structures→ Computational geometry; Mathematics of computing→ Discrete mathematics → Graph theory → Graph algorithms; Mathematics of computing → Discrete mathematics → Graph theory → Graphs and surfaces; Mathematics of computing → Continuous mathematics → Topology → Algebraic topology
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