At the core of the seminal Graph Minor Theory of Robertson and Seymour is a powerful structural theorem capturing the structure of graphs excluding a fixed minor. This result is used throughout graph theory and graph algorithms, but is existential. We develop a polynomialtime algorithm using topological graph theory to decompose a graph into the structure guaranteed by the theorem: a clique-sum of pieces almost-embeddable into boundedgenus surfaces. This result has many applications. In particular, we show applications to developing many approximation algorithms, including a 2-approximation to graph coloring, constant-factor approximations to treewidth and the largest grid minor, combinatorial polylogarithmicapproximation to half-integral multicommodity flow, subexponential fixed-parameter algorithms, and PTASs for many minimization and maximization problems, on graphs excluding a fixed minor. 1.

In 1977, Wegner conjectured that the chromatic number of the square of every planar graph G with maximum degree ∆ ≥ 8 is at most ⌊ 3 2 ∆ ⌋ + 1. We show that it is at most 3 2 ∆ (1 + o(1)), and indeed this is true for the list chromatic number and for more general classes of graphs.

Suppose G is a graph of bounded degree d, and one needs to remove ɛn of its edges in order to make it planar. We show that in this case the statistics of local neighborhoods around vertices of G is far from the statistics of local neighborhoods around vertices of any planar graph G ′. In fact, a similar result is proved for any minor-closed property of bounded degree graphs. As an immediate corollary of the above result we infer that many well studied graph properties, like being planar, outer-planar, series-parallel, bounded genus, bounded tree-width and several others, are testable with a constant number of queries. None of these properties was previously known to be testable even with o(n) queries. 1

Let H be a fixed graph. We show that any H-minor free graph of high enough girth has a circular-chromatic number arbitrarily close to two. Equivalently, such graphs have homomorphisms into a large odd circuit. In particular, graphs of high girth and of bounded genus or bounded tree width are "nearly bipartite" in this sense. For example, any planar graph of girth at least 16 admits a homomorphism onto a pentagon. We also obtain tight bounds in a few specific cases of small forbidden minors.

this paper is to reduce the upper bound for the required girth to the correct order of magnitude: Theorem 1. For any inte k,e ve graph G of girth g(G) > 6 log k +3and #(G) # 3 has a minor H with #(H) >k. The best lower bound we have found is 8 3 log c, but we note that existing conjectures about cubic graphs of large girth would raise this to about 4 log . Since an average degree of at least cr # log r forces a K r minor [ 5, 8 ], Theorem 1 has the following consequence: Corollary 2.The ee a constant c # R such thate ea graph G of girth g(G) # 6 log r<F1

Abstract. According to Euler’s formula, every planar graph with n vertices has at most O(n) edges. How much can we relax the condition of planarity without violating the conclusion? After surveying some classical and recent results of this kind, we prove that every graph of n vertices, which can be drawn in the plane without three pairwise crossing edges, has at most O(n) edges. For straight-line drawings, this statement has been established by Agarwal et al., using a more complicated argument, but for the general case previously no bound better than O(n 3/2) was known. 1

Let K ∗ s,t denote the graph obtained from Ks,t by adding all edges between the s vertices of degree t in it. We show how to adapt the argument of an our previous paper (Discrete Math. 308 (2008), 4435–4445) to prove that if t / log 2 t ≥ 1000s, then every graph G with average degree at least t + 8s log 2 s has a K ∗ s,t minor. This refines a corresponding result by Kühn and Osthus. AMS Subject Classification: 05C35, 05C83.

Let G be an n-vertex m-edge graph with weighted vertices. A pair of vertex sets A, B ⊆ V (G) is a 2/3

Abstract. We improve the approximation ratios for two optimization problems in planar graphs. For node-weighted Steiner tree, a classical network-optimization problem, the best achievable approximation ratio in general graphs is Θ(log n), and nothing better was previously known for planar graphs. We give a constant-factor approximation for planar graphs. Our algorithm generalizes to allow as input any nontrivial minor-closed graph family, and also generalizes to address other optimization problems such as Steiner forest, prize-collecting Steiner tree, and network-formation games. The second problem we address is group Steiner tree: given a graph with edge weights and a collection of groups (subsets of nodes), find a minimum-weight connected subgraph that includes at least one node from each group. The best approximation ratio known in general graphs is O(log 3 n), or O(log 2 n) when the host graph is a tree. We obtain an O(log n polyloglog n) approximation algorithm for the special case where the graph is planar embedded and each group is the set of nodes on a face. We obtain the same approximation ratio for the minimum-weight tour that must visit each group. 1

Abstract. There is substantial literature dealing with fixed parameter algorithms for the dominating set problem on various families of graphs. In this paper, we give a k O(dk) n time algorithm for finding a dominating set of size at most k in a d-degenerated graph with n vertices. This proves that the dominating set problem is fixed-parameter tractable for degenerated graphs. For graphs that do not contain Kh as a topological minor, we give an improved algorithm for the problem with running time (O(h)) hk n. For graphs which are Kh-minor-free, the running time is further reduced to (O(log h)) hk/2 n. Fixed-parameter tractable algorithms that are linear in the number of vertices of the graph were previously known only for planar graphs. For the families of graphs discussed above, the problem of finding an induced cycle of a given length is also addressed. For every fixed H and k, we show that if an H-minor-free graph G with n vertices contains an induced cycle of size k, then such a cycle can be found in O(n) expected time as well as in O(n log n) worst-case time. Some results are stated concerning the (im)possibility of establishing linear time algorithms for the more general family of degenerated graphs. Keywords: H-minor-free graphs, degenerated graphs, dominating set problem, finding an induced cycle, fixed-parameter tractable algorithms. 1