Approximating the list-chromatic number and the chromatic number in minor-closed and odd-minor-closed classes of graphs

It is well-known (Feige and Kilian [24], H̊astad [39]) that approximating the chromatic number within a factor of n1−ε cannot be done in polynomial time for ε> 0, unless coRP = NP. Computing the list-chromatic number is much harder than determining the chromatic number. It is known that the problem of deciding if the list-chromatic number is k, where k ≥ 3, is Πp2-complete [37]. In this paper, we focus on minor-closed and odd-minor-closed families of graphs. In doing that, we may as well consider only graphs without Kk-minors and graphs with-out odd Kk-minors for a fixed value of k, respectively. Our main results are that there is a polynomial time approxi-mation algorithm for the list-chromatic number of graphs without Kk-minors and there is a polynomial time approxi-mation algorithm for the chromatic number of graphs with-out odd-Kk-minors. Their time complexity is O(n 3) and O(n4), respectively. The algorithms have multiplicative er-ror O( log k) and additive error O(k), and the multiplica-tive error occurs only for graphs whose list-chromatic num-ber and chromatic number are Θ(k), respectively. Let us recall that H has an odd complete minor of order l if there are l vertex disjoint trees in H such that every two of them are joined by an edge, and in addition, all the vertices ∗Research partly supported by Japan Society for the Pro

Structure Theorem and Isomorphism Test for Graphs with Excluded Topological Subgraphs

We generalize the structure theorem of Robertson and Seymour for graphs excluding a fixed graph $H$ as a minor to graphs excluding $H$ as a topological subgraph. We prove that for a fixed $H$, every graph excluding $H$ as a topological subgraph has a tree decomposition where each part is either "almost embeddable" to a fixed surface or has bounded degree with the exception of a bounded number of vertices. Furthermore, we prove that such a decomposition is computable by an algorithm that is fixed-parameter tractable with parameter $|H|$. We present two algorithmic applications of our structure theorem. To illustrate the mechanics of a "typical" application of the structure theorem, we show that on graphs excluding $H$ as a topological subgraph, Partial Dominating Set (find $k$ vertices whose closed neighborhood has maximum size) can be solved in time $f(H,k)\cdot n^{O(1)}$ time. More significantly, we show that on graphs excluding $H$ as a topological subgraph, Graph Isomorphism can be solved in time $n^{f(H)}$. This result unifies and generalizes two previously known important polynomial-time solvable cases of Graph Isomorphism: bounded-degree graphs and $H$-minor free graphs. The proof of this result needs a generalization of our structure theorem to the context of invariant treelike decomposition