The Oriented Diameter of Graphs with Given Connected Domination Number and Distance Domination Number
Peter Dankelmann, J.B. MORGAN, Emily Rivett-Carnac · Graphs and Combinatorics · 2024
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 $$\gamma _c(G)$$ 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 $$2 \gamma _c(G) +3$$ 2 γ c ( G ) + 3 if $$\gamma _c(G)$$ γ c ( G ) is even and $$2 \gamma _c(G) +2$$ 2 γ c ( G ) + 2 if $$\gamma _c(G)$$ γ c ( G ) is odd. This bound is sharp. For $$d \in {\mathbb {N}}$$ d ∈ N , the d-distance domination number $$\gamma ^d(G)$$ 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 $$(2d+1)(d+1)\gamma ^d(G)+ O(d)$$ ( 2 d + 1 ) ( d + 1 ) γ d ( G ) + O ( d ) . Furthermore, we construct bridgeless graphs whose oriented diameter is at least $$(d+1)^2 \gamma ^d(G) +O(d)$$ ( d + 1 ) 2 γ d ( G ) + O ( d ) , thus demonstrating that our above bound is best possible apart from a factor of about 2.