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- Peter Dankelmann
- Discrete Applied Mathematics
- 1997

- Peter Dankelmann, David P. Day, David Erwin, Simon Mukwembi, Henda C. Swart
- Discrete Mathematics
- 2009

Let G be a graph and S ⊆ V (G). For each vertex u ∈ S and for each v ∈ V (G) − S, we define d(u, v) = d(v, u) to be the length of a shortest path in V (G)−(S−{u}) if such a path exists, and ∞ otherwise. Let v ∈ V (G). We define w S (v) = u∈S 1 2 d(u,v)−1 if v ∈ S, and w S (v) = 2 if v ∈ S. If, for each v ∈ V (G), we have w S (v) ≥ 1, then S is an… (More)

- Peter Dankelmann
- Discrete Applied Mathematics
- 1994

A sharp upper bound on the average distance of a graph depending on the order and the independence number is given. As a corollary we obtain the maximum average distance of a graph with given order and matching number. All extremal graphs are determined.

- Peter Dankelmann, I. Gutman, Simon Mukwembi, Henda C. Swart
- Discrete Mathematics
- 2009

If G is a connected graph, then the distance between two edges is, by definition, the distance between the corresponding vertices of the line graph of G. The edge-Wiener index W e of G is then equal to the sum of distances between all pairs of edges of G. We give bounds on W e in terms of order and size. In particular we prove the asymptotically sharp upper… (More)

- Peter Dankelmann
- Inf. Process. Lett.
- 1993

An O(m)-algorithm for computing the average distance of an interval graph with m edges of unit length is presented. A slight modification of the algorithm can be used for computing the average distance of a tree with weighted edges in time O(n), where n is the number of vertices.

- Elias Dahlhaus, Peter Dankelmann, Wayne Goddard, Henda C. Swart
- Discrete Applied Mathematics
- 2003

For a graph G with weight function w on the vertices, the total distance of G is the sum over all unordered pairs of vertices x and y of w(x)w(y) times the distance between x and y. A MAD tree of G is a spanning tree with minimum total distance. We develop a linear-time algorithm to find a MAD tree of a distance-hereditary graph; that is, those graphs where… (More)

- Peter Dankelmann, David P. Day, Johannes H. Hattingh, Michael A. Henning, Lisa R. Markus, Henda C. Swart
- Discrete Mathematics
- 2007

Let G = (V, E) be a graph. A set S ⊆ V is a restrained dominating set if every vertex not in S is adjacent to a vertex in S and to a vertex in V \ S. The restrained domination number of G, denoted by γ r (G), is the minimum cardinality of a restrained dominating set of G. A set S ⊆ V is a total dominating set if every vertex in V is adjacent to a vertex in… (More)

- Elias Dahlhaus, Peter Dankelmann, R. Ravi
- Inf. Process. Lett.
- 2004

The average distance of a connected graph G is the average of the distances between all pairs of vertices of G. We present a linear time algorithm that determines, for a given interval graph G, a spanning tree of G with minimum average distance (MAD tree). Such a tree is sometimes referred to as a minimum routing cost spanning tree.

The eccentricity of a vertex is the maximum distance from it to another vertex and the average eccentricity of a graph is the mean eccentricity of a vertex. In this paper we establish bounds on the mean eccentricity of a graph. We then examine the change in the average eccentricity when a graph is replaced by a spanning subgraph, in particular the two… (More)

- Peter Dankelmann, Roger C. Entringer
- Journal of Graph Theory
- 2000