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An {\em orientation} of an undirected graph $G$ is an assignment of exactly one direction to each edge of $G$. Converting two-way traffic networks to one-way traffic networks and bidirectional communication networks to unidirectional communication networks are practical instances of graph orientations. In these contexts minimising the diameter of the resulting oriented graph is of prime interest. The $n$-star network topology was proposed as an alternative to the hypercube network topology for multiprocessor systems by Akers and Krishnamurthy [IEEE Trans. on Computers (1989)]. The $n$-star graph $S_n$ consists of $n!$ vertices, each labelled with a distinct permutation of $[n]$. Two vertices are adjacent if their labels differ exactly in the first and one other position. $S_n$ is an $(n-1)$-regular, vertex-transitive graph with diameter $\lfloor 3(n-1)/2 \rfloor$. Orientations of $S_n$, called unidirectional star graphs and distributed routing protocols over them were studied by Day and Tripathi [Information Processing Letters (1993)] and Fujita [The First International Symposium on Computing and Networking (CANDAR 2013)]. Fujita showed that the (directed) diameter of this unidirectional star graph $\overrightarrow{S_n}$ is at most $\lceil{5n/2}\rceil + 2$. In this paper, we propose a new distributed routing algorithm for the same $\overrightarrow{S_n}$ analysed by Fujita, which routes a packet from any node $s$ to any node $t$ at an undirected distance $d$ from $s$ using at most $\min\{4d+4, 2n+4\}$ hops. This shows that the (directed) diameter of $\overrightarrow{S_n}$ is at most $2n+4$. We also show that the diameter of $\overrightarrow{S_n}$ is at least $2n$ when $n \geq 7$, thereby showing that our upper bound is tight up to an additive factor., Comment: Full version of the paper to be presented in CALDAM 2020

An $n$-tournament $T$ with vertex set $V$ is simple if there is no subset $M$ of $V$ such that $2\leq \left \vert M\right \vert \leq n-1$ and for every $x\in V\setminus M$, either $M\rightarrow x$ or $x \rightarrow M$. The simplicity index of an $n$-tournament $T$ is the minimum number $s(T)$ of arcs whose reversal yields a non-simple tournament. M\"{u}ller and Pelant (1974) proved that $s(T)\leq\frac{n-1}{2}$, and that equality holds if and only if $T$ is doubly regular. As doubly regular tournaments exist only if $n\equiv 3\pmod{4}$, $s(T)<\frac{n-1}{2}$ for $n\not\equiv3\pmod{4}$. In this paper, we study the class of $n$-tournaments with maximal simplicity index for $n\not\equiv3\pmod{4}$.

We give a quasi-complete solution of the (\Delta,N) problem for two well-known families of digraphs used as good models for large interconnection networks. In our study we also relate both families, the New Amsterdam and Manhattan digraphs, with the double-step graphs (or circulant graphs with degree two).