Barát and Thomassen (2006) posed the following decomposition conjecture: for each tree T , there exists a natural number kT such that, if G is a kT -edge-connected graph and |E(G)| is divisible by |E(T )|, then G admits a decomposition into copies of T . In a series of papers, Thomassen verified this conjecture for stars, some bistars, paths of length 3, and paths whose length is a power of 2. ...

We consider straight-line outerplanar drawings of outerplanar graphs in which a small number of distinct edge slopes are used, that is, the segments representing edges are parallel to a small number of directions. We prove that ∆ − 1 edge slopes suffice for every outerplanar graph with maximum degree ∆ > 4. This improves on the previous bound of O(∆), which was shown for planar partial 3-trees,...

A forest in which every component is path is caned a path forest. A family of path forests whose edge sets form a partition of the set of a graph G is called a path of G. The minimum number of forests in a path decomposition of a graph G is the path number of G and denoted by p( G). If we restrict the number of edges in each path to be at most x then we obtain a special decomposition. The minim...

We derive precise asymptotic estimates for the number of labelled graphs not containing K3,3 as a minor, and also for those which are edge maximal. Additionally, we establish limit laws for parameters in random K3,3-minor-free graphs, like the expected number of edges. To establish these results, we translate a decomposition for the corresponding graph class into equations for generating functi...

Dynamic programming is a method in which the solution to a computational problem is found by combination of already obtained solutions to subproblems. This method can be applied to problems on graphs (nodes connected by edges). The graph of interest must then be broken down into successively smaller parts according to some suitable principle. The thesis studies two graph algebras introduced for...

A decomposition of a graph G is a set D = {H1, · · · , Hk} of pairwise edge-disjoint subgraphs of G that cover the set of edges of G. If Hi is isomorphic to a fixed graph H, for 1 ≤ i ≤ k, then we say that D is an H-decomposition of G. In this work, we study the case where H is a path of fixed length. For that, we first decompose the given graph into trails, and then we use a disentangling lemm...

We derive precise asymptotic estimates for the number of labelled graphs not containing K3,3 as a minor, and also for those which are edge maximal. Additionally, we establish limit laws for parameters in random K3,3-minor-free graphs, like the number of edges. To establish these results, we translate a decomposition for the corresponding graphs into equations for generating functions and use si...