We present a necessary and sufficient condition for a graph of odd-girth 2k + 1 to bound the class of K4-minor-free graphs of odd-girth (at least) 2k + 1, that is, to admit a homomorphism from any such K4-minor-free graph. This yields a polynomial-time algorithm to recognize such bounds. Using this condition, we first prove that every K4-minor free graph of odd-girth 2k+1 admits a homomorphism ...

A list-colouring of a graph is an assignment of a colour to each vertex v from its own list L(v) of colours. Instead of colouring vertices we may want to colour other elements of a graph such as edges, faces, or any combination of vertices, edges and faces. In this thesis we will study several of these different types of list-colouring, each for the class of a near-outerplanar graphs. Since a g...

In the course of extending Grötzsch’s theorem, we prove that every triangle-free graph without a K5-minor is 3-colorable. It has been recently proved that every triangle-free planar graph admits a homomorphism to the Clebsch graph. We also extend this result to the class of triangle-free graphs without a K5-minor. This is related to some conjectures which generalize the Four-Color Theorem. Whil...

A subgraph-universal graph/a topological minor-universal graph in a class of graphs G ${\mathscr{G}}$ is which contains every as subgraph/topological minor. We prove that the P ${\mathscr{P}}$ all countable planar does not contain graph. This answers question Diestel and Kühn strengthens result Pach stating there no . Furthermore, we characterise for subdivided stars T $T$ -free graphs.

An infinite graph Γ is minor excluded if there is a finite graph that is not a minor of Γ. We prove that minor excluded graphs have finite Assouad-Nagata dimension and study minor exclusion for Cayley graphs of finitely generated groups. Our main results and observations are: (1) minor exclusion is not a group property: it depends on the choice of generating set; (2) a group with one end has a ...

1 PROBLEM DEFINITION The theory of bidimensionality provides general techniques for designing efficient fixed-parameter algorithms and approximation algorithms for a broad range of NP-hard graph problems in a broad range of graphs. This theory applies to graph problems that are " bidi-mensional " in the sense that (1) the solution value for the k ×k grid graph and similar graphs grows with k, t...

We show that every K4-minor free graph with toughness greater than 4/7 has a 2-walk, i.e., a closed walk visiting each vertex at most twice. We also give an example of a 4/7-tough K4-minor free graph with no 2-walk.

The Induced Minor problem is that of testing whether a graph G can be modified into a graph H by a sequence of vertex deletions and edge contractions. If only edge contractions are permitted, we obtain the Contractibility problem. We prove that Induced Minor is polynomial-time solvable when G is AT-free and H is fixed, i.e., not part of the input. In addition, we show that Contractiblity is pol...

We establish the complexity of several graph embedding problems: Subgraph Isomorphism, Graph Minor, Induced Subgraph and Induced Minor, when restricted to H-minor free graphs. In each of these problems, we are given a pattern graph P and a host graph G, and want to determine whether P is a subgraph (minor, induced subgraph or induced minor) of G. We show that, for any fixed graph H and > 0, if ...

We prove that any H-minor-free graph, for a fixed graph H, of treewidth w has an Ω(w)× Ω(w) grid graph as a minor. Thus grid minors suffice to certify that H-minor-free graphs have large treewidth, up to constant factors. This strong relationship was previously known for the special cases of planar graphs and bounded-genus graphs, and is known not to hold for general graphs. The approach of thi...