Clin d’oeil on Lj-embeddable planar graphs

Clin D'oeil on L1-embeddable Planar Graphs

Random graphs on surfaces

Counting labelled planar graphs, and typical properties of random labelled planar graphs, have received much attention recently. We start the process here of extending these investigations to graphs embeddable on any fixed surface S. In particular we show that the labelled graphs embeddable on S have the same growth constant as for planar graphs, and the same holds for unlabelled graphs. Also, ...

Counting planar graphs and related families of graphs

In this article we survey recent results on the asymptotic enumeration of planar graphs and, more generally, graphs embeddable in a fixed surface and graphs defined in terms of excluded minors. We also discuss in detail properties of random planar graphs, such as the number of edges, the degree distribution or the size of the largest k-connected component. Most of the results we present use gen...

Constant-Factor Approximations for Asymmetric TSP on Nearly-Embeddable Graphs

In the Asymmetric Traveling Salesperson Problem (ATSP) the goal is to find a closed walk of minimum cost in a directed graph visiting every vertex. We consider the approximability of ATSP on topologically restricted graphs. It has been shown by Oveis Gharan and Saberi [14] that there exists polynomial-time constant-factor approximations on planar graphs and more generally graphs of constant ori...

On the Maximum Degree of a Random Planar Graph

Let the random graph Rn be drawn uniformly at random from the set of all simple planar graphs on n labelled vertices. We see that with high probability the maximum degree of Rn is Θ(lnn). We consider also the maximum size of a face and the maximum increase in the number of components on deleting a vertex. These results extend to graphs embeddable on any fixed surface.