The parameter contraction degeneracy — the maximum minimum degree over all minors of a graph — is a treewidth lower bound and was first defined in [3]. In experiments it was shown that this lower bound improves upon other treewidth lower bounds [3]. In this note, we examine some relationships between the contraction degeneracy and connected components of a graph, blocks of a graph and the genus of a graph. We also look at chordal graphs, and we study an upper bound on the contraction degeneracy. A data structure that can be used for algorithms computing the degeneracy and similar parameters, is also described.

We address in this paper the problem of constructing embeddings of planar graphs satisfying declarative, user-defined topological constraints. The constraints consist each of a cycle...

All existing algorithms for planarity testing/planar embedding can be grouped into two principal classes. Either, they run in linear time, but to the expense of complex algorithmic concepts or complex data-structures, or they are easy to understand and implement, but require more than linear time [2]. In this paper, a new linear-time algorithm for embedding maximal planar graphs is proposed. This algorithm is both easy to understand and easy to implement. The algorithm consists of three phases which use only simple, local graph-modifications. In addition to planar embedding, the new algorithm allows to test graphs for maximal planarity. The generation of Straight Line Drawings by a technique of Read [12] can be naturally incorporated into the algorithm. We also demonstrate how to generate random (maximal) planar graphs. The algorithm presented constitutes a first step towards a simple, linear-time solution for embedding general planar graphs. 2 3 1 Introduction One of the main top...

. This paper concerns graph embedding under topological constraints. We address the problem of finding a planar embedding of a graph satisfying a set of constraints between its vertices and cycles that require embedding a given vertex inside its corresponding cycle. This problem turns out to be NP-complete. However, towards an analysis of its tractable subproblems, we develop an efficient algorithm for the special case where graphs are 2-connected and any two distinct cycles in the constraints have at most one vertex in common. 1 Introduction In many graph drawing applications it is important to find drawings with certain geometrical or topological properties. For instance, to support the semantics of visual languages, given subgraphs should be drawn with a predefined shape. Such user-defined requirements on the graph layout are called graph drawing constraints. Due to their wide applicability, graph drawing approaches supporting them are of recent research interest [7]. So far most o...

In Chapter 1 we introduced the puzzle of the three houses and the three utilities. The problem was to determine if we could connect each of the three utilities with each of the three houses so that none of the utility lines crossed. We attempted a drawing of the graph model K 3, 3 in Figure 1.1.2. In this chapter we will show that no such drawing is possible in the plane. A (p, q) graph G is said to be embeddable in the plane or planar if it is possible to draw G in the plane so that the edges of G intersect only at end vertices. If such a drawing has been done, we say that a plane embedding of the graph has been found. v 1 v 2 r 1 r 2 r 3 v 3 v 4 r 4 Figure 6.1.1. A plane embedding of K 4. Given a plane graph G, a region of G is a maximal section of the plane for which any two points may be joined by a curve. Intuitively, a region is the connected section of the plane bounded (often enclosed) by some set of edges of G. In Figure 6.1.1, the regions of G are labeled r 1, r 2, r 3 and r 4. The region r 1 is bounded by the edges v 1 v 2, v 2 v 3 and v 3 v 1, while r 2 is bounded by v 1 v 2, v 2 v 4 and v 4 v 1. Further, note that every plane graph has a region similar to the region r 1. This region, which is actually not enclosed, is called the exterior region. It is also the case that given any planar graph G, there is an embedding of G with any region as the exterior region. One of the most useful results for dealing with planar graphs relates the order, size and number of regions of the graph. This result is originally from Euler [1].

A graph is planar if and only if it can be embedded in the plane without crossings. I give a detailed exposition of simple and efficient, yet poorly known algorithms for planarity testing, embedding, and Kuratowski subgraph extraction based on the left-right characterization of planarity.