On the Wiener Index of Orientations of Graphs

The Wiener index of a strong digraph $D$ 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 $a$ to a vertex $b$ as $0$ if there is no path from $a$ to $b$ in $D$. 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 $T$, an orientation $D$ of $T$ of maximum Wiener index always contains a vertex $v$ such that for every vertex $u$, there is either a $(u,v)$-path or a $(v,u)$-path in $D$. 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 $m$ edges has an orientation of Wiener index $m$ can be solved in time quadratic in $n$.