A totally odd H-subdivision means a subdivision of a graph H in which each edge of H corresponds to a path of odd length. Thus this concept is a generalization of a subdivision of H. In this paper, we give a structure theorem for graphs without a fixed graph H as a totally odd subdivision. Namely, every graph with no totally odd H-subdivision has a tree-decomposition such that each piece is eit...

We design approximation algorithms for Unique Games when the constraint graph admits good low diameter graph decomposition. For the Max-2Link problem in Kr-minor free graphs, when there is an assignment satisfying 1 − ε fraction of constraints, we present an algorithm that produces an assignment satisfying 1−O(rε) fraction of constraints, with the approximation ratio independent of the alphabet...

Motivated by previous results on distance constrained labelings and coloring of squares of K4-minor free graphs, we show that for every p ≥ q ≥ 1, there exists ∆0 such that every K4-minor free graph G with maximum degree ∆ ≥ ∆0 has an L(p, q)-labeling of span at most qb3∆(G)/2c. The obtained bound is the best possible.

A minor-closed class of graphs is a set of labelled graphs which is closed under isomorphism and under taking minors. For a minor-closed class G, we let gn be the number of graphs in G which have n vertices. A recent result of Norine et al. [12] shows that for all minor-closed class G, there is a constant c such that gn ≤ c nn!. Our main results show that the growth rate of gn is far from arbit...

Free-minor closed classes [2] and free-planar graphs [3] are considered. Versions of Kuratowski-like theorem for free-planar graphs and Kuratowski theorem for planar graphs are considered. We are using usual definitions of the graph theory [1]. Considering graph topologically and Kuratowski theorem, we use the notion of minor following the theory of Robertson and Seymour[2]. We say, that a grap...

Robin Thomas asked whether for every proper minor-closed class G, there exists a polynomial-time algorithm approximating the chromatic number of graphs from G up to a constant additive error independent on the class G. We show this is not the case: unless P = NP, for every integer k ≥ 1, there is no polynomial-time algorithm to color a K4k+1-minor-free graph G using at most χ(G) + k − 1 colors....

We give polynomial-time constant-factor approximation algorithms for the treewidth and branchwidth of any H-minor-free graph for a given graph H with crossing number at most 1. The approximation factors are 1.5 for treewidth and 2.25 for branchwidth. In particular, our result directly applies to classes of nonplanar graphs such as K5-minorfree graphs and K3,3-minor-free graphs. Along the way, w...

4 A 3-connected graph is 3-connected if it has no 3-separation that separates a “large” fan or K3,n 5 from the rest of the graph. It is proved in this paper that, except for K4, every 3 -connected graph has 6 a 3-connected proper minor that is at most two edges away from the original graph. This result is used 7 to characterize Q-minor-free graphs, where Q is obtained from the Cube by contracti...