Frank Göring

Embedded in the Shadow of the Separator

Eigenvectors to the second smallest eigenvalue of the Laplace matrix of a graph, also known as Fiedler vectors, are the basic ingredient in spectral graph partitioning heuristics. Maximizing this second smallest eigenvalue over all nonnegative edge weightings with bounded total weight yields the absolute algebraic connectivity introduced by Fiedler, who proved tight connections of this value to the connectivity of the graph. Our objective is to gain a better understanding of the connections between separators and the eigenspace of this eigenvalue by studying the dual semidefinite optimization problem to the absolute algebraic connectivity. By exploiting optimality conditions we show that this problem is equivalent to finding an embedding of the

The rotational dimension of a graph

n

Graph realizations associated with minimizing the maximum eigenvalue of the Laplacian

nodes of the graph in

ON F-INDEPENDENCE IN GRAPHS

n

Short proof of Menger's Theorem

-space so that their barycenter is the origin, the distance between adjacent nodes is bounded by one, and the nodes are spread as much as possible (the sum of the squared norms is maximized). For connected graphs we prove that, for any separator in the graph, at least one of the two separated node sets is embedded in the shadow (with the origin being the light source) of the convex hull of the separator. Furthermore, we show that there always exists an optimal embedding whose dimension is bounded by the tree width of the graph plus one.

GreedyMAX-type algorithms for the maximum independent set problem

Given a connected graph G = (N, E) with node weights s∈ℝ **image** and nonnegative edge lengths, we study the following embedding problem related to an eigenvalue optimization problem over the second smallest eigenvalue of the (scaled) Laplacian of G: Find vi∈ℝ|N|, i∈N so that distances between adjacent nodes do not exceed prescribed edge lengths, the weighted barycenter of all points is at the origin, and **image** is maximized. In the case of a two-dimensional optimal solution this corresponds to the equilibrium position of a quickly rotating net consisting of weighted mass points that are linked by massless cables of given lengths. We define the rotational dimension of G to be the minimal dimension k so that for all choices of lengths and weights an optimal solution can be found in ℝk and show that this is a minor monotone graph parameter. We give forbidden minor characterizations up to rotational dimension 2 and prove that the rotational dimension is always bounded above by the tree-width of G plus one.

Locally Dense Independent Sets in Regular Graphs of Large Girth—An Example of a New Approach

In analogy to the absolute algebraic connectivity of Fiedler, we study the problem of minimizing the maximum eigenvalue of the Laplacian of a graph by redistributing the edge weights. Via semidefinite duality this leads to a graph realization problem in which nodes should be placed as close as possible to the origin while adjacent nodes must keep a distance of at least one. We prove three main results for a slightly generalized form of this embedding problem. First, given a set of vertices partitioning the graph into several or just one part, the barycenter of each part is embedded on the same side of the affine hull of the set as the origin. Second, there is an optimal realization of dimension at most the tree-width of the graph plus one and this bound is best possible in general. Finally, bipartite graphs possess a one dimensional optimal embedding.

On domination in graphs

Let F be a set of graphs and for a graph G let F(G) and F (G) denote the maximum order of an induced subgraph of G which does not contain a graph in F as a subgraph and which does not contain a graph in F as an induced subgraph, respectively. Lower bounds on F(G) and F (G) are presented.

Unique-Maximum Coloring Of Plane Graphs

Abstract A short proof of the classical theorem of Menger concerning the number of disjoint AB -paths of a finite digraph for two subsets A and B of its vertex set is given.

On cycles through specified vertices

A maximum independent set problem for a simple graph G = (V, E) is to find the largest subset of pairwise nonadjacent vertices. The problem is known to be NP-hard and it is also hard to approximate. Within this article we introduce a non-negative integer valued function p defined on the vertex set V (G) and called a potential function of a graph G, while P(G) = maxv∈V(G) p(v) is called a potential of G. For any graph P(G) ≤ Δ(G), where Δ(G) is the maximum degree of G. Moreover, Δ(G) - P(G) may be arbitrarily large. A potential of a vertex lets us get a closer insight into the properties of its neighborhood which leads to the definition of the family of GreedyMAX-type algorithms having the classical GreedyMAX algorithm as their origin. We establish a lower bound 1/(P + 1) for the performance ratio of GreedyMAX-type algorithms which favorably compares with the bound 1/(Δ + 1) known to hold for GreedyMAX. The cardinality of an independent set generated by any GreedyMAX-type algorithm is at least Σv∈V(G)(p(v)+1)-1, which strengthens the bounds of Turaan and Caro-Wei stated in terms of vertex degrees.