We provide a complete structural characterization of K2,4-minor-free graphs. The 3-connected K2,4minor-free graphs consist of nine small graphs on at most eight vertices, together with a family of planar graphs that contains K4 and, for each n ≥ 5, 2n − 8 nonisomorphic graphs of order n. To describe the 2-connected K2,4-minor-free graphs we use xy-outerplanar graphs, graphs embeddable in the pl...

The clustered chromatic number of a class of graphs is the minimum integer k such that for some integer c every graph in the class is k-colourable with monochromatic components of size at most c. We prove that for every graph H , the clustered chromatic number of the class of H-minor-free graphs is tied to the tree-depth of H . In particular, if H has tree-depth t then every H-minor-free graph ...

Hadwiger’s conjecture states that every graph with chromatic number χ has a clique minor of size χ. Let G be a graph on n vertices with chromatic number χ and stability number α. Then since χα ≥ n, Hadwiger’s conjecture implies that G has a clique minor of size n α . In this paper we prove that this is true for connected claw-free graphs with α ≥ 3. We also show that this result is tight by pro...

The value of depth-first search or "backtracking" as a technique for solving graph problems is illustrated by two examples. An algorithm for finding the biconnected components of an undirected graph and an improved version of an algorithm for finding the strongly connected components of a directed graph are presented. The space and time requirements of both algorithIns are bounded by kl V+ k2E+...

We study max-cut in classes of graphs defined by forbidding a single graph as a subgraph, induced subgraph, or minor. For the first two containment relations, we prove dichotomy theorems. For the minor order, we show how to solve max-cut in polynomial time for the class obtained by forbidding a graph with crossing number at most one (this generalizes a known result for K5-minor-free graphs) and...

Gluck [6] has proven that triangulated 2-spheres are generically 3-rigid. Equivalently, planar graphs are generically 3-stress free. We show that already the K5-minor freeness guarantees the stress freeness. More generally, we prove that every Kr+2-minor free graph is generically r-stress free for 1 ≤ r ≤ 4. (This assertion is false for r ≥ 6.) Some further extensions are discussed.

Tutte showed that 4-connected planar graphs are Hamiltonian, but it is well known that 3-connected planar graphs need not be Hamiltonian. We show that K2,5-minor-free 3-connected planar graphs are Hamiltonian. This does not extend to K2,5-minor-free 3-connected graphs in general, as shown by the Petersen graph, and does not extend to K2,6-minor-free 3-connected planar graphs, as we show by an i...

We introduce a new framework for designing fixed-parameter algorithms with subexponential running time—2 √ n. Our results apply to a broad family of graph problems, called bidimensional problems, which includes many domination and covering problems such as vertex cover, feedback vertex set, minimum maximal matching, dominating set, edge dominating set, clique-transversal set, and many others re...

The spectral radius ρ G of a graph G is the largest eigenvalue of its adjacency matrix. Let λ G be the smallest eigenvalue of G. In this paper, we have described the K3,3-minor free graphs and showed that A let G be a simple graph with order n ≥ 7. If G has no K3,3-minor, then ρ G ≤ 1 √3n − 8. B LetG be a simple connected graph with order n ≥ 3. IfG has noK3,3-minor, then λ G ≥ −√2n − 4, where ...

We give the first linear kernels for Dominating Set and Connected Dominating Set problems on graphs excluding a fixed graph H as a minor. In other words, we give polynomial time algorithms that, for a given H-minor free graph G and positive integer k, output an H-minor free graph G′ on O(k) vertices such that G has a (connected) dominating set of size k if and only if G′ has. Prior to our work,...