3 research outputs found
On the Wiener Index of Orientations of Graphs
The Wiener index of a strong digraph is defined as the sum of the
distances between all ordered pairs of vertices. This definition has been
extended to digraphs that are not necessarily strong by defining the distance
from a vertex to a vertex as if there is no path from to in
.
Knor, \u{S}krekovski and Tepeh [Some remarks on Wiener index of oriented
graphs. Appl.\ Math.\ Comput.\ {\bf 273}] considered orientations of graphs
with maximum Wiener index. The authors conjectured that for a given tree ,
an orientation of of maximum Wiener index always contains a vertex
such that for every vertex , there is either a -path or a
-path in . In this paper we disprove the conjecture.
We also show that the problem of finding an orientation of maximum Wiener
index of a given graph is NP-complete, thus answering a question by Knor,
\u{S}krekovski and Tepeh [Orientations of graphs with maximum Wiener index.
Discrete Appl.\ Math.\ 211].
We briefly discuss the corresponding problem of finding an orientation of
minimum Wiener index of a given graph, and show that the special case of
deciding if a given graph on edges has an orientation of Wiener index
can be solved in time quadratic in