Abstract
AbstractLet 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.
Funder
National Research Foundation
Ernest Oppenheimer Memorial Trust
Publisher
Springer Science and Business Media LLC
Subject
Discrete Mathematics and Combinatorics,Theoretical Computer Science