On Oriented Diameter of $(n, k)$-Star Graphs

Abstract: Assignment of one of the two possible directions to every edge of an undirected graph $G=(V,E)$ is called an orientation of $G$. The resulting directed graph is denoted by $\overrightarrow{G}$. A strong orientation is one in which every vertex is reachable from every other vertex via a directed path. The diameter of $\overrightarrow{G}$, i.e., the maximum distance from one vertex to another, depends on the particular orientation. The minimum diameter among all possible orientations is called the oriented diameter $\overrightarrow{\text{diam}}(G)$ of $G$. Let $n,k$ be two integers with $1 \leq k < n$. In the realm of interconnection networks of processing elements, an $(n,k)$-star graph $S_{n,k}$ offers a topology that circumvents the lack of scalability of $n$-star graphs $S_n$. In this paper, we present a strong orientation for $S_{n,k}$ that combines approaches suggested by Cheng and Lipman [Journal of Interconnection Networks (2002)] for $S_{n,k}$ with the one proposed by Fujita [The First International Symposium on Computing and Networking (CANDAR 2013)] for $S_n$. Next, we propose a distributed routing algorithm for $\overrightarrow{S_{n,k}}$ inspired by an algorithm proposed by Kumar, Rajendraprasad and Sudeep [Discrete Applied Mathematics (2021)] for $\overrightarrow{S_n}$. With the aid of both the orientation scheme and the routing algorithm, we show that $\overrightarrow{\text{diam}}(S_{n,k}) \leq \lfloor \frac{n+k}{2} \rfloor + 2k + 6 - \delta(n,k)$ where $\delta(n,k)$ is a non-negative function. The function $\delta(n,k)$ takes on values $2k-n$, $0$, and $\left\lfloor \frac{n-3k}{2} \right\rfloor$ respectively for three disjoint intervals $k>\frac{n}{2}$, $\frac{n}{3} < k \leq \frac{n}{2}$ and $k\leq \frac{n}{3}$. For every value of $n$, $k$, our upper bound performs better than all known bounds in literature.