Nathaniel Dean

Bounds for rectilinear crossing numbers

A rectilinear drawing of a graph is one where each edge is drawn as a straight-line segment, and the rectilinear crossing number of a graph is the minimum number of crossings over all rectilinear drawings. We describe, for every integer k ≥ 4, a class of graphs of crossing number k, but unbounded rectilinear crossing number. This is best possible since the rectilinear crossing number is equal to the crossing number whenever the latter is at most 3. Further, if we consider drawings where each edge is drawn as a polygonal line segment with at most one break point, then the resulting crossing number is at most quadratic in the regular crossing number.

The matching extendability of surfaces

A connected graph G having at least 2n + 2 vertices is said to be n-extendable if it contains a matching of size n and every such matching is contained in a perfect matching. M. D. Plummer posed the problem of determining the smallest integer μ(Σ) such that no graph embeddable in the surface Σ is μ(Σ)-extendable. We call μ(Σ) the matching extendability of Σ and show that if Σ is not homeomorphic to the sphere then μ(Σ) = 2 + ⌊ 4 − χ ⌋ where χ is the Euler characteristic of Σ. In particular, no projective planar graph is 3-extendable.

Graph Drawing and Manipulation with LINK

This paper introduces the LINK system as a flexible tool for the creation, manipulation, and drawing of graphs and hypergraphs. We describe the basic architecture of the system and illustrate its flexibility with several examples. LINK is distinguished from existing software for discrete mathematics by its layered interface, including a graphical user interface tied into an object-oriented Scheme language interface with access to Tk, and an extensible underlying set of C++ libraries. We conclude by briefly discussing roles LINK has played in research and education.

Well-covered graphs and extendability

Abstract A graph is k-extendable if every independent set of size k is contained in a maximum independent set. This generalizes the concept of a B-graph (i.e. 1-extendable graph) introduced by Berge and the concept of a well-covered graph (i.e. k-extendable for every integer k) introduced by Plummer. For various graph families we present some characterizations of well-covered and k-extendable graphs. We show that in order to determine whether a graph is well-covered it is sometimes sufficient to verify that it is k-extendable for small values of k. For many classes of graphs, this leads to efficient algorithms for recognizing well-covered graphs.

On obstructions to small face covers in planar graphs

Abstract Several algorithmic and graph-theoretic developments have focused on the problem of covering, in a planar graph, selected vertices with fewest possible faces. This paper discusses some obstructions to the existence of small face covers. If the embedding of the graph is fixed, this problem leads to variants of the Erdos-Posa theorem on independent cycles in a graph. If the embedding of the graph is not fixed, the analysis leads to generalizations of outerplanar graphs, and we obtain an explicit upper bound on the size of the minimal excluded minors for such classes of graphs.

Distribution of contractible edges in k -connected graphs

Abstract An edge xy of a k-connected graph G is said to be k-contractible if the graph G · xy obtained from G by contracting xy is k-connected. We derive several new results on the distribution of k-contractible edges. Let G[Ek(G)] be subgraph of G induced by the set Ek(G) of k-contractible edges in G. We show that if G is a k-connected graph (k ≥ 2) which is triangle-free or has minimum degree at least ⌊ 3k 2 ⌋, then G[Ek(G)] is 2-connected and spans G. Furthermore, if k ≥ 3, then G contains an induced cycle C such that every edge of C is k-contractible and G − V(C) is connected.

Gallai's conjecture for disconnected graphs

Abstract The path number p ( G ) of a graph G is the minimum number of paths needed to partition the edge set of G. Gallai conjectured that p ( G )⩽⌊( n +1)/2⌋ for every connected graph G of order n. Because the graph consisted of disjoint triangles, the best one could hope for in the disconnected case is p(G)⩽⌊ 2 3 n⌋ . We prove the sharper result that p(G)⩽ 1 2 u+⌊ 2 3 g⌋ where u is the number of odd vertices and g is the number of nonisolated even vertices.

New results on rectilinear crossing numbers and plane embeddings

We show that if a graph has maximum degree d and crossing number k, its rectilinear crossing number is at most O(dk2). Hence for graphs of bounded degree, the crossing number and the rectilinear crossing number are bounded as functions of one another. We also obtain a generalization of Tuttes theorem on convex embeddings of 3-connected plane graphs.

Cycles of length 0 modulo 4 in graphs

Abstract In several papers a variety of questions have been raised concerning the existence of cycles of length 0 mod k in graphs. For the case k =4, we answer three of these questions by showing that a graph G contains such a cycle provided it has any of the following three properties: (1) G has minimum degree at least 2 and at most two vertices of degree 2, (2) G is not 3-colorable, and (3) G is a subdivision of a graph of order p ⩾5 with at least 3 p -5 edges.

Mathematical programming model of bond length and angular resolution for minimum energy carbon nanotubes

This paper addresses the problem of determining minimum energy configurations of single-walled carbon nanotubes through the use of a mathematical programming model. The model includes a potential energy function which is minimized subject to constraints on the angular resolution and bond lengths. This approach seems to consistently produce stable configurations.