The Oriented Diameter of Graphs with Given Connected Domination Number and Distance Domination Number

Let G be a bridgeless graph. An orientation of G is a digraph obtained from G by assigning a direction to each edge. The oriented diameter of G is the minimum diameter among all strong orientations of G. The connected domination number γ_c(G) of G is the minimum cardinality of a set S of vertices of G such that every vertex of G is in S or adjacent to some vertex of S, and which induces a connected subgraph in G. We prove that the oriented diameter of a bridgeless graph G is at most 2 γ_c(G) + 3 if γ_c(G) is even and 2 γ_c(G) + 2 if γ_c(G) is odd. This bound is sharp. For d∈ N , the d-distance domination number γ^d(G) of G is the minimum cardinality of a set S of vertices of G such that every vertex of G is at distance at most d from some vertex of S. As an application of a generalisation of the above result on the connected domination number, we prove an upper bound on the oriented diameter of the form (2 d+ 1) (d+ 1) γ^d(G) + O(d) . Furthermore, we construct bridgeless graphs whose oriented diameter is at least (d+ 1) ²γ^d(G) + O(d) , thus demonstrating that our above bound is best possible apart from a factor of about 2.