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

Abstract

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 \(\gamma _c(G)\) 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 \(2 \gamma _c(G) +3\) if \(\gamma _c(G)\) is even and \(2 \gamma _c(G) +2\) if \(\gamma _c(G)\) is odd. This bound is sharp. For \(d \in {\mathbb {N}}\), the d-distance domination number \(\gamma ^d(G)\) 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 \((2d+1)(d+1)\gamma ^d(G)+ O(d)\). Furthermore, we construct bridgeless graphs whose oriented diameter is at least \((d+1)^2 \gamma ^d(G) +O(d)\), thus demonstrating that our above bound is best possible apart from a factor of about 2.