Abstract
It is known that embedding a graph G into a surface of minimum genus γmin(G) is NP-hard, whereas embedding a graph G into a surface of maximum genus γ m(G) can be done in polynomial time. However, the complexity of embedding a graph G into a surface of genus between γ min(G) and γ m(G) is still unknown. In this paper, it is proved that for any function f(n)=O(n) ∈, 0 ≤ε<1, the problem of embedding a graph G of n vertices into a surface of genus at most γ min(G)+f(n) remains NP-hard, while there is a linear time algorithm that approximates the minimum genus embedding either within a constant ratio or within a difference O(n). A polynomial time algorithm is also presented for embedding a graph G into a surface of genus γ m(G)−1.
Supported by the National Sciecne Foundation under Grant CCR-9110824.
Supported by Engineering Excellence Award from Texas A&M University.
Supported in part by the NATO Scientific Affairs Division under collaborative research grant 911016.
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© 1993 Springer-Verlag Berlin Heidelberg
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Chen, J., Kanchi, S.P., Kanevsky, A. (1993). On the complexity of graph embeddings. In: Dehne, F., Sack, JR., Santoro, N., Whitesides, S. (eds) Algorithms and Data Structures. WADS 1993. Lecture Notes in Computer Science, vol 709. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-57155-8_251
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DOI: https://doi.org/10.1007/3-540-57155-8_251
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