We prove two structural decomposition theorems about graphs excluding a fixed odd minor H, and show how these theorems can be used to obtain approximation algorithms for several algorithmic problems in such graphs. Our decomposition results provide new structural insights into odd-H-minor-free graphs, on the one hand generalizing the central structural result from Graph Minor Theory, and on the other hand providing an algorithmic decomposition into two bounded-treewidth graphs, generalizing a

more »

... milar result for minors. As one example of how these structural results conquer difficult problems, we obtain a polynomial-time 2-approximation for vertex coloring in odd-H-minor-free graphs, improving on the previous O(|V (H)|)-approximation for such graphs and generalizing the previous 2-approximation for H-minor-free graphs. The class of odd-H-minor-free graphs is a vast generalization of the wellstudied H-minor-free graph families and includes, for example, all bipartite graphs plus a bounded number of apices. Odd-H-minorfree graphs are particularly interesting from a structural graph theory perspective because they break away from the sparsity of Hminor-free graphs, permitting a quadratic number of edges. position (and thus its algorithmic consequences) cannot be generalized to odd-minor-free graphs, because small separators do not exist. Another contrast between odd-Hminor-free graphs and H-minor-free graphs is that the former can have a quadratic number of edges, while the latter are always sparse. Odd-minor-free graphs have been considered extensively in the graph theory literature (see, e.g., [Gue01, GGG + 04, Gue05, JT95]) and recently in theoretical computer science [KM06] . We prove our result by generalizing a structural result that is the heart of Graph Minor Theory [RS03], which has many algorithmic applications [Gro03, DFHT05, DH05, DHK05], to the case of odd-minor-free graphs. Specifically, we prove that every odd-minor-free graph can be decomposed into a clique-sum of "almost-bipartite" graphs and graphs that are "almost-embeddable" into bounded-genus surfaces. (In the original Graph Minor version, only graphs of the second type exist.) Our primary challenge in this result is that odd-minor-free graphs can be dense, and this density may be equally spread throughout the graph, so we cannot hope to find small separations to split the graph into pieces as in previous decomposition theorems. Nonetheless, we show how to decompose odd-minor-free graphs, by showing a connection to the existence of clique minors; this algorithmic technique may be useful for future structural algorithms on dense graphs. All of our results are algorithmic: the decompositions can be computed in polynomial time. Our decompositions can also be used to obtain a variety of approximation algorithms for problems on odd-minor-free graphs. In particular, our decomposition into two bounded-treewidth graphs (Theorem 1.1 below) immediately leads to a 2-approximation for many NP-complete problems in odd-minor-free graphs. For example, our 2-approximation for graph coloring in odd-K kminor-free graphs improves the recent O(k)-approximation algorithm [KM06] and generalizes the 2-approximation for K k -minor-free graphs [DHK05] . We also believe that combining our clique-sum decomposition (Theorem 1.2 below) with Grohe's technique [Gro03] leads to PTASs for maximum independent set, maximum clique, and minimum vertex cover. Grohe's technique for minor-free graphs uses the property that removing small sets of vertices in such graphs results in components with treewidth bounded in terms of diameter; this property simply for odd-minor-free graphs (e.g., the complete bipartite graph has diameter 2 but huge treewidth even after removing small sets of vertices), making this generalization particularly interesting. The classes of graphs excluding a fixed minor H have already proved extremely useful and powerful, with applications in many areas such as network design and compact routing; see, e.g., [AG06, Tho04] . Our thesis is that most algorithmic results that have been obtained for both H-minor-free graphs and bipartite graphs can be general-