Dieter Rautenbach

Wiener index versus maximum degree in trees

The Wiener index of a graph is the sum of all pairwise distances of vertices of the graph. In this paper, we characterize the trees which minimize the Wiener index among all trees of given order and maximum degree and the trees which maximize the Wiener index among all trees of given order that have only vertices of two different degrees.

On the Randić index

The Randic index R(G) of a graph G = (V, E) is the sum of (d(u)d(υ))-1/2 over all edges uυ ∈ E of G. Bollobas and Erdos (Ars Combin. 50 (1998) 225) proved that the Randic index of a graph of order n without isolated vertices is at least √n - 1. They asked for the minimum value of R(G) for graphs G with given minimum degree δ(G). We answer their question for δ(G) = 2 and propose a related conjecture. Furthermore, we prove a best-possible lower bound on the Randic index of a triangle-free graph G with given minimum degree δ(G).

On the Band-, Tree-, and Clique-Width of Graphs with Bounded Vertex Degree

The band-, tree-, and clique-width are of primary importance in algorithmic graph theory due to the fact that many problems that are NP-hard for general graphs can be solved in polynomial time when restricted to graphs where one of these parameters is bounded. It is known that for any fixed

Some results on graphs without long induced paths

\Delta \geq 3

Extremal Chemical Trees

, all three parameters are unbounded for graphs with vertex degree at most

A generalization of Dijkstra's shortest path algorithm with applications to VLSI routing

\Delta

Fundamental limits on the anonymity provided by the MIX technique

. In this paper, we distinguish representative subclasses of graphs with bounded vertex degree that have bounded band-, tree-, or clique-width. Our proofs are constructive and lead to efficient algorithms for a variety of NP-hard graph problems when restricted to those classes.

New bounds on the k-domination number and the k-tuple domination number

Many NP-hard graph problems remain difficult on Pk-free graphs for certain values of k. Our goal is to distinguish subclasses of Pk-free graphs where several important graph problems can be solved in polynomial time. In particular, we show that the independent set problem is polynomial-time solvable in the class of (Pk, K1,n)-free graphs for any positive integers k and n, thereby generalizing several known results.

Efficient Dominating and Edge Dominating Sets for Graphs and Hypergraphs

Avariety of molecular-graph-based structure-descriptors were proposed, in particular the Wiener index W, the largest graph eigenvalue λ1, the connectivity index χ, the graph energy E and the Hosoya index Z, capable of measuring the branching of the carbon-atom skeleton of organic compounds, and therefore suitable for describing several of their physico-chemical properties. We now determine the structure of the chemical trees (= the graph representation of acyclic saturated hydrocarbons) that are extremal with respect to W, λ1, E, and Z, whereas the analogous problem for χ was solved earlier. Among chemical trees with 5, 6, 7, and 3k + 2 vertices, k = 2, 3,..., one and the same tree has maximum λ1 and minimum W, E, Z. Among chemical trees with 3k and 3k + 1 vertices, k = 3, 4..., one tree has minimum W and maximum λ1 and another minimum E and Z.

Some remarks on the geodetic number of a graph

We generalize Dijkstras algorithm for finding shortest paths in digraphs with non-negative integral edge lengths. Instead of labeling individual vertices we label subgraphs which partition the given graph. We can achieve much better running times if the number of involved subgraphs is small compared to the order of the original graph and the shortest path problems restricted to these subgraphs is computationally easy.As an application we consider the VLSI routing problem, where we need to find millions of shortest paths in partial grid graphs with billions of vertices. Here, our algorithm can be applied twice, once in a coarse abstraction (where the labeled subgraphs are rectangles), and once in a detailed model (where the labeled subgraphs are intervals). Using the result of the first algorithm to speed up the second one via goal-oriented techniques leads to considerably reduced running time. We illustrate this with a state-of-the-art routing tool on leading-edge industrial chips.