Oriented diameter of graphs with given domination number

Abstract

Let $G$ be a connected bridgeless graph with domination number $\gamma$. The oriented diameter (strong diameter) of $G$ is the smallest integer $d$ for which $G$ admits a strong orientation with diameter (strong diameter) $d$. Kurz and L\"atsch (2012) conjectured the oriented diameter of $G$ is at most $\lceil \frac{7\gamma+1}{2}\rceil$ and the bound is sharp. In this paper, we confirm the conjecture by induction on $\gamma$ through contracting an unavoidable alternative subgraph, which holds potential for future applications. Moreover, we show the oriented strong diameter of $G$ is at most $7\gamma-1$ by using the same recursive structure, and the bound is best possible.