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 (2d+1)(d+1)γ ^d(G)+ O(d) . Furthermore, we construct bridgeless graphs whose oriented diameter is at least (d+1)^2 γ ^d(G) +O(d) , thus demonstrating that our above bound is best possible apart from a factor of about 2.