For a fixed finite family of graphs F, the F-Minor-Free Deletion problem takes as input graph G and integer ℓ asks whether size-ℓ vertex set X exists such that G−X is F-minor-free. {K2}-Minor-Free {K3}-Minor-Free encode Vertex Cover Feedback Set respectively. When parameterized by feedback number these two problems are known to admit polynomial kernelization. We show {P3}-Minor-Free MK[2]-hard....

We propose local versions of Hadwiger's Conjecture, where only balls radius Ω(log⁡(v(G))) around each vertex are required to be Kt-minor-free. ask: if a graph is locally-Kt-minor-free, it t-colourable? show that the answer yes when t≤5, even in stronger setting list-colouring, and we complement this result with O(log⁡v(G))-round distributed colouring algorithm LOCAL model. Further, for large en...

We develop new structural results for apex-minor-free graphs and show their power by developing two new approximation algorithms. The first is an additive approximation for coloring within 2 of the optimal chromatic number, which is essentially best possible, and generalizes the seminal result by Thomassen [32] for bounded-genus graphs. This result also improves our understanding from an algori...

Thomassen proved that all planar graphs are 5-choosable. Škrekovski strengthened the result by showing K5-minor-free Dvo?ák and Postle pointed out DP-5-colorable. In this note, we first improve these results every or K3,3-minor-free graph is final section, further under term strictly f-degenerate transversal.

In his classic paper Über eine Eigenschaft der ebenen Komplexe, Wagner [ 19 ] tackles the following problem. Kuratowski’s theorem, in its excluded minor version, states that a finite graph is planar if and only if it has no minor isomorphic to K5 or to K3,3. (A minor of G is any graph obtained from some H ⊂ G by contracting connected subgraphs.) If we exclude only one of these two minors, the g...

Let G be a K4-minor-free graph with maximum degree . It is known that if ∈ {2, 3} then G2 is ( + 2)-degenerate, so that (G2) ch(G2) + 3. It is also known that if 4 then G2 is ( 3 2 + 1)-degenerate and (G2) 3 2 + 1. It is proved here that if 4 then G2 is 3 2 -degenerate and ch(G2) 3 2 + 1. These results are sharp. © 2007 Elsevier B.V. All rights reserved.

Hadwiger’s conjecture states that every graph with chromatic number χ has a clique minor of size χ. In this paper we prove a weakened version of this conjecture for the class of claw-free graphs (graphs that do not have a vertex with three pairwise nonadjacent neighbors). Our main result is that a claw-free graph with chromatic number χ has a clique minor of size ⌈23χ⌉.

A key theorem in algorithmic graph-minor theory is a min-max relation between the treewidth of a graph and its largest grid minor. This min-max relation is a keystone of the Graph Minor Theory of Robertson and Seymour, which ultimately proves Wagner’s Conjecture about the structure of minor-closed graph properties. In 2008, Demaine and Hajiaghayi proved a remarkable linear min-max relation for ...

A signed graph is a graph where each edge is labeled as either positive or negative. A circle is positive if the product of edge labels is positive. The frustration index is the least number of edges that need to be removed so that every remaining circle is positive. The maximum frustration of a graph is the maximum frustration index over all possible sign labellings. We prove two results about...

It is proved that, if G is a K4-minor-free graph with maximum degree ∆ > 4, then G is totally (∆ + 1)-choosable; that is, if every element (vertex or edge) of G is assigned a list of ∆ + 1 colours, then every element can be coloured with a colour from its own list in such a way that every two adjacent or incident elements are coloured with different colours. Together with other known results, t...