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Fundamental Cycles and Graph Embeddings

By Ren Han, Zhao Hongtao and Li Haoling


In this paper, we investigate fundamental cycles in a graph G and their relations with graph embeddings. We show that a graph G may be embedded in an orientable surface with genus at least g if and only if for any spanning tree T, there exists a sequence of fundamental cycles C1, C2, · · ·, C2g with C2i−1 ∩ C2i ̸ = φ for 1 ≤ i ≤ g. In particular, among β(G) fundamental cycles of any spanning tree T of a graph G, there are exactly 2γM(G) cycles C1, C2, · · ·, C2γM(G) such that C2i−1 ∩ C2i ̸ = φ for 1 ≤ i ≤ γM(G), where β(G) and γM(G) are, respectively, the Betti number and the maximum genus of G. This implies that it is possible to construct an orientable embedding with large genus of a graph G from an arbitrary spanning tree T ( which may have very large number of odd components in G\E(T)). This is different from the earlier work of Xuong and Liu[9,6], where spanning trees with small odd components are needed. In fact, this makes a common generalization of Xuong[9],Liu[6] and Fu et al[2]. Further more, we show that (1).This result is useful in locating the maximum genus of a graph having a specific edge-cut. Some known results for embedded graphs are also concluded;(2).The maximum genus problem may be reduced to the maximum matching problem. Based on this result and the algorithm of Micali-Vazirani[8], we present a new efficient algorithm to determine the maximum genus of a graph in O((β(G)) 5 2) steps. Our method is straight and quite deferent from the algorithm of Furst,Gross and McGeoch[3] which depends on a result of Giles[4]where matroid parity method is needed

Topics: Fundamental cycles, Maximum genus, upper-embedded. AMS 2000
Year: 2008
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