@TECHREPORT{Hardy05oddcycles, author = {Nadia Hardy}, title = {Odd cycles in planar graphs}, institution = {}, year = {2005} }

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Abstract

Given a graph G = (V,E), an odd cycle cover is a subset of the vertices whose removal makes the graph bipartite, that is, it meets all odd cycles in G. A packing in G is a collection of vertex disjoint odd cycles. This thesis addresses algorithmic and structural problems concerning odd cycle covers and packings. In particular, we consider the two NP-hard problems of finding a maximum packing and a minimum covering. In 1994 Brass [53] conjectured that τ, the minimum size of an odd cycle cover, is at most twice ν, the maximum size of a packing. The conjecture is known to be false in general [11, 41]. We prove here that τ ≤ 10ν for planar graphs. Our structural results leads to the first constant approximation algorithm for the packing problem. The covering problem was shown to be tractable for graphs of constant sized solutions [42]. We give a linear time algorithm for the covering problem restricted to the case where the graphs have constant sized solutions and are planar.