NDLI: Fundamental cycles and graph embeddings

Content Provider	Springer Nature Link
Author	Ren, Han Zhao, HongTao Li, HaoLing
Copyright Year	2009
Abstract	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 C $_{1}$,C $_{2}$,…,C $_{2g }$ with C $_{2i−1}$ ∩ C $_{2i }$ ≠ /0 for 1 ⩽ i ⩽ g. In particular, among β(G) fundamental cycles of any spanning tree T of a graph G, there are exactly 2$_{γM }$(G) cycles C $_{1}$, C $_{2}$,…,C $_{2γM(G)}$ such that C $_{2i−1}$ ∩ C $_{2i }$ ≠ /0 for 1 ⩽ i ⩽ $_{γM }$(G), where β(G) and $_{γM }$(G) are the Betti number and the maximum genus of G, respectively. 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, where spanning trees with small odd components are needed. In fact, this makes a common generalization of Xuong, Liu and Fu et al. Furthermore, we show that (1) this result is useful for 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, we present a new efficient algorithm to determine the maximum genus of a graph in $$ O((\beta (G))^{\frac{5} {2}} ) $$ steps. Our method is straight and quite different from the algorithm of Furst, Gross and McGeoch which depends on a result of Giles where matroid parity method is needed.
Starting Page	1920
Ending Page	1926
Page Count	7
File Format	PDF
ISSN	10069283
Journal	Science in China Series A: Mathematics
Volume Number	52
Issue Number	9
e-ISSN	18622763
Language	English
Publisher	SP Science in China Press
Publisher Date	2009-09-03
Publisher Place	Heidelberg
Access Restriction	Subscribed
Subject Keyword	fundamental cycle maximum genus upper-embedded Planar graphs; geometric and topological aspects of graph theory Factorization, matching, partitioning, covering and packing Applications of Mathematics
Content Type	Text
Resource Type	Article
Subject	Mathematics

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