Oriented diameter of graphs with given domination number

Let G be a connected bridgeless graph with domination number $γ$. 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 Lätsch (2012) conjectured the oriented diameter of G is at most 7γ+1/2 and the bound is sharp. In this paper, we confirm the conjecture by induction on $γ$ through contracting an unavoidable alternative subgraph, which holds potential for future applications. Moreover, we show the oriented strong diameter of G is at most $7γ-1$ by using the same recursive structure, and the bound is best possible.