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.

1 Introduction

The distance between two vertices u and v in a graph (or digraph) G, denoted by $d_G(u,v)$, is the length of a shortest path from u to v in G. The maximum of all the distances between two vertices in a graph (or digraph) G is the diameter of G and is denoted by ${\textrm{diam}}(G)$.

An orientation of a graph G is a digraph obtained from G by assigning a direction to each edge. We often denote an orientation of G by $\overrightarrow{G}$. We say that an orientation is strong if there is a path from every vertex to every other vertex in $\overrightarrow{G}$. The $\textit{oriented diameter}$ of G is the smallest diameter among all strong orientations of G and is denoted by $\overrightarrow{{\textrm{diam}}}(G)$.

A bridge of a connected graph is an edge whose removal disconnects the graph. A connected graph containing no bridges is said to be bridgeless. A well-known result due to Robbins [17] states that a graph admits a strong orientation, if and only if it is bridgeless. However, it does not give any information on how the distances increase in such an orientation.

Clearly, the oriented diameter of any bridgeless graph is at least two. Chvátal and Thomassen [4] proved that deciding if a given graph has an orientation of diameter two, and thus determining the oriented diameter, is NP-complete. Sufficient conditions for a graph to have an orientation of diameter two are thus of interest. Czabarka et al. [7] gave a degree condition on the minimum degree that guarantees an orientation of diameter two. Later, Chen and Chang [3] considered bipartite graphs and gave a sufficient condition in terms of the minimum degree for such graphs to have oriented diameter three. Cochran et al. [6] proved that for $n \ge 5$, every simple graph of order n and size at least $\left( {\begin{array}{c}n\\ 2\end{array}}\right) -n+5$ has an orientation of diameter two, thus proving a conjecture by Koh and Tay [13].

Chvátal and Thomassen [4] also bounded the oriented diameter of G from above by $2{\textrm{diam}}(G)^2 + 2{\textrm{diam}}(G)$, and for every $d \ge 2$ constructed bridgeless graphs of diameter d in which every orientation has diameter at least $\frac{1}{2}d^2 + d$. Recently, Babu et al. [1] improved on Chvátal and Thomassen’s [4] upper bound to $1.373 {\textrm{diam}}(G)^2 + 6.971 {\textrm{diam}}(G)-1$, which outperforms the previous upper bound for values of ${\textrm{diam}}(G)$ greater than or equal to 8.

Bounds on the oriented diameter in terms of other graph invariants and graph classes have been investigated, including minimum degree [2, 5, 18], maximum degree [8], chordal graphs [11], maximal outerplanar graphs [19] and planar triangulations [15].

Building on earlier work by Koh and Tay [12], the oriented diameter of graphs obtained by replacing the vertices of a given graph by independent sets of given order has recently been investigated in [16, 20,21,22,23].

In this paper we explore upper bounds on the oriented diameter in terms of domination-type parameters. A set S of vertices of a graph G is called a dominating set of G if every vertex in $V(G)-S$ is adjacent to some vertex of S. The domination number $\gamma (G)$ of a graph G is the minimum number of vertices in a dominating set.

Fomin, Matamala, Prisner and Rapaport [10] proved that the oriented diameter cannot exceed $9\gamma (G)-5$, and this was improved to $5 \gamma (G) -1$ in [9]. Kurz and Lätsch [14] gave the currently strongest upper bound in terms of the domination number, $4 \gamma (G)$, and conjectured that the minimum oriented diameter for bridgeless graphs is at most $\lceil \frac{7 \gamma (G) +1}{2}\rceil$. If true, this bound is sharp.

In this paper we present an upper bound on the oriented diameter of a bridgeless graph of given connected domination number $\gamma _c(G)$, defined as the minimum cardinality of a dominating set that induces a connected subgraph. Unlike the above results on the domination number, our bound is sharp.

For $d\in {\mathbb {N}}$, the d-distance domination number of G, denoted by $\gamma ^d(G)$, is the minimum cardinality of a set S of vertices of G such that every vertex of G is within distance d from some vertex of S. We prove the upper bound $(2d+1)(d+1) \gamma ^d(G) +O(d)$ on the oriented diameter of a bridgeless graph in terms of its d-distance domination number and show that for all d this bound is sharp apart from a factor of at most about 2.

We obtain this bound as a corollary to a bound on the oriented diameter in terms of a generalisation of the connected domination number, the connected d-distance domination number of G, defined as the minimum cardinality of a set S of vertices of G such that S induces a connected graph in G, and every vertex of G is within distance d of some vertex of S.

This paper is organised as follows. In Sect. 2 we fix the terminology and notation for this paper. In Sect. 3 we introduce k-extensions. These are used in Sect. 4 to prove a sharp upper bound on the oriented diameter in terms of the connected domination number. A more general bound in terms of the connected d-distance domination number is obtained in Sect. 5, from which we derive a bound on the oriented diameter in terms of the d-distance domination number in Sect. 6.

2 Terminology and Notation

The notation we use is as follows. $G=(V,E)$ denotes a finite connected graph or multigraph with vertex set V and edge set E, which we denote by V(G) and E(G), respectively. By the order of the graph we mean $\vert V(G) \vert$. An edge between two vertices u and v is denoted by $uv \in E(G)$ and an edge that is oriented from u to v is denoted by $\overrightarrow{uv}$. By the neighbourhood $N_G(v)$ of a vertex v of G, we mean the set of vertices of G that are adjacent to v. The degree of a vertex v is the number of edges incident with v. An end-vertex is a vertex of degree one. For a digraph $\overrightarrow{G}$, a vertex u is an in-neighbour of $v \in V(\overrightarrow{G})$ if $\overrightarrow{uv} \in E(\overrightarrow{G})$ and an out-neighbour if $\overrightarrow{vu} \in E(\overrightarrow{G})$.

A path Q from u to v will be referred to as a (u, v)-path and its length is denoted by $\ell (Q)$. For a set X of vertices of G, we define $d_G(u,X)=\min _{x \in X} d_G(u,x)$. The eccentricity ${\textrm{ecc}}_G(v)$ of a vertex v in G is the distance from v to a vertex farthest from v. The minimum of the eccentricities of the vertices of G is the radius of G. For a digraph $\overrightarrow{G}$, the out-eccentricity of a vertex v of $\overrightarrow{G}$ is the greatest distance from v to a vertex $u \in V(\overrightarrow{G})$. The in-eccentricity of a vertex v of $\overrightarrow{G}$ is the greatest distance from a vertex $u \in V(\overrightarrow{G})$ to v. The eccentricity ${\textrm{ecc}}_{\overrightarrow{G}}$ of a vertex v of $\overrightarrow{G}$ is the maximum of its out-eccentricity and in-eccentricity. The radius of $\overrightarrow{G}$ is the minimum of the eccentricities of its vertices. The oriented radius of an undirected graph G, denoted by $\overrightarrow{{\textrm{rad}}}(G)$, is the smallest radius among all strong orientations of G.

A graph G is bipartite if its vertex set can be partitioned into two sets $V_1$ and $V_2$ such that every edge joins a vertex in $V_1$ to a vertex in $V_2$; these two sets are referred to as the partite sets of G. The union $G_1 \cup G_2$ of graphs $G_1=(V_1, E_1)$ and $G_2=(V_2, E_2)$ is the graph with vertex set $V_1 \cup V_2$ and edge set $E_1 \cup E_2$. If S is a set of vertices of G, then the subgraph induced by S, denoted by G[S], is the subgraph of G with vertex set S, in which two vertices are adjacent if and only if they are adjacent in G. A cut-vertex is a vertex of a connected graph whose removal disconnects the graph.

3 k-Extensions

In this section we prove a lemma which is the main tool for the proofs of our main results. In order to state the lemma, we require two definitions.

Def 1

Let T be a not necessarily spanning subtree of a multigraph G. By a T-path we mean a path in $G-E(T)$ whose ends are distinct vertices of T, but whose internal vertices are not. We say that a T-path P covers an edge e of T if e is on the unique cycle contained in $T\cup P$. By a T-cycle we mean a cycle in $G-E(T)$ that shares exactly one vertex with T. If P is a T-path or a T-cycle, then we refer to the vertices of P that are not in T (that are in T) as the internal vertices (the ends) of P.

If T is a tree and ${{{\mathcal {P}}}}$ is a collection of T-paths and T-cycles such that every edge of T is covered by some path in ${{{\mathcal {P}}}}$, then clearly $T \cup \bigcup _{P \in {{{\mathcal {P}}}}} P$ is a bridgeless graph. We note that the converse is not necessarily true.

Def 2

Let T be a subtree of a multigraph G. Let ${{{\mathcal {P}}}}$ be a set of (not necessarily disjoint) T-paths and T-cycles, and let $k \in {\mathbb {N}}$. We say that ${{{\mathcal {P}}}}$ is a k-extension of T in G if no T-path or T-cycle in ${{{\mathcal {P}}}}$ has length greater than k, and $T \cup \bigcup _{P \in {{{\mathcal {P}}}}} P$ is bridgeless.

Lemma 1

Let G be a multigraph and T a (not necessarily spanning) subtree of G of order p, where $p\ge 1$. Let ${{{\mathcal {P}}}}$ be a k-extension of T, $k\ge 1$, and

$H = T \cup \bigcup _{P \in {{{\mathcal {P}}}}} P$. Then there exists a strong orientation A of some submultigraph of H containing T such that

• (i) for every two vertices $u,v \in V(T)$,

  $$\begin{aligned} d_A(u,v) \le {\left\{ \begin{array}{ll} \frac{k+1}{2}p -1 &{} \quad \text {for }p\text { even,} \\ \frac{k+1}{2}(p-1) &{} \quad \text {for }p\text { odd}, \\ \end{array}\right. } \end{aligned}$$

  (1)

• (ii) for every two vertices $u,v \in V(A)$,

  $$\begin{aligned} d_A(u,v) \le {\left\{ \begin{array}{ll} \frac{k+1}{2}p + k-2 &{} \quad \text {for }p\text { even,} \\ \frac{k+1}{2}p + \frac{k-3}{2} &{} \quad \text {for }p\text { odd}. \\ \end{array}\right. } \end{aligned}$$

  (2)

Proof

The statement clearly holds for $k=1$ since $T \cup \bigcup _{P \in {{{\mathcal {P}}}}} P$ is a bridgeless multigraph of order p containing T, and the diameter of a strong orientation is at most $p-1$. Hence, we assume that $k \ge 2$.

We prove the statement by induction on p. The statement clearly holds for $p=1$ since we can choose A to contain only the vertex of T and no edge. So we assume that $p \ge 2$ and that the statement holds for all trees of order less than p.

Case 1: There exists a T-path $P \in {{{\mathcal {P}}}}$ that covers more than one edge of T.

For $u, v \in V(T)$, let u and v be the ends of P on T and let $d_T(u,v)=l$, so $l \ge 2$. Let C be the unique cycle in $T \cup P$. In G, T and in all T-paths and T-cycles of ${{\mathcal {P}}}$ identify all the vertices on C to a new vertex $u'$. This may create multiple edges; however, we delete any loops that arise.

Let $G'$ and $T'$ be the multigraph obtained from G and T, respectively. Clearly, $T'$ is a tree. We obtain a k-extension ${{{\mathcal {P}}}}'$ of $T'$ from ${{{\mathcal {P}}}}$ as follows. Let $Q \in {{{\mathcal {P}}}}$ be a T-path or a T-cycle not containing an internal vertex that is also in C. Then Q gives rise to a either a $T'$-path in $G'$ (if at most one end of Q is a vertex of C) or a $T'$-cycle in $G'$ (if both ends of Q are vertices of C), which we call $Q'$. In either case replace Q by $Q'$. Now let $Q \in {{{\mathcal {P}}}}$ be a T-path or a T-cycle containing an internal vertex that is in C. Then Q gives rise to a walk $Q'$ in $G'$, in which no vertex except possibly $u'$ is repeated. If x and y are the ends of $Q'$ then let $Q' =xQ_1'u' Q_2'u',\ldots ,u'Q_s'y$, where each $Q_i'$ is a segment of $Q'$ between two consecutive appearances of $u'$, or between $u'$ and an end of $Q'$. Clearly, each $Q_i'$ is either a $T'$-path or a $T'$-cycle. We replace Q by $Q_1', Q_2',\ldots ,Q_s'$. It is easy to verify that ${{{\mathcal {P}}}}'$ is a k-extension of $T'$.

Let $H'=T'\ \cup \bigcup _{P \in {{{\mathcal {P}}}'}} P$ and $\vert V(T') \vert = p-l$. By the induction hypothesis there exists an orientation $A'$ of a submultigraph of $H'$ containing $T'$ such that

• (i) for any $a,b \in V(T')$,

  $$\begin{aligned} d_{A'}(a,b) \le {\left\{ \begin{array}{ll} \frac{k+1}{2}(p-l) -1 &{} \quad \text {for }p-l\text { even,} \\ \frac{k+1}{2}(p-l-1) &{} \quad \text {for }p=l\text { odd}, \\ \end{array}\right. } \end{aligned}$$

  (3)

• (ii) for any $a,b \in V(A')$,

  $$\begin{aligned} d_{A'}(a,b) \le {\left\{ \begin{array}{ll} \frac{k+1}{2}(p-l) + k-2 &{} \quad \text {for }p-l\text { even,} \\ \frac{k+1}{2}(p-l) + \frac{k-3}{2} &{}\quad \text {for }p-l\text { odd}. \\ \end{array}\right. } \end{aligned}$$

  (4)

We now obtain the orientation A of a submultigraph of H from $A'$ as follows. All edges in $A'$ not incident with $u'$ retain their orientation in A and the edges of C get the orientation of a directed cycle. For $u \in V(C)$, an edge uv of some $P \in {{{\mathcal {P}}}}$ (that after the contraction becomes $u'v$ of some $P' \in {{{\mathcal {P}}}}'$) gets the same orientation as in $A'$ i.e., if $u'v$ is oriented as $\overrightarrow{u'v}$ ($\overrightarrow{vu'}$) in $A'$, then uv is oriented as $\overrightarrow{uv}$ ($\overrightarrow{vu}$) in A. Let $a,b \in V(G)$. Let $a'=a$ if $a \notin V(C)$ and $a'=u'$ if $a \in V(C)$. Define $b'$ analogously. We show that a directed $(a',b')$-path $R'$ in $A'$ gives rise to a directed (a, b)-path $\overrightarrow{R}$ in A with $\ell (\overrightarrow{R}) \le \ell (R') + |V(C)|-1$. Indeed, if $R'$ does not contain $u'$, then let $\overrightarrow{R}=R'$, and if $R'$ contains $u'$, then replacing $u'$ by a suitable segment of the directed cycle $\overrightarrow{C}$ yields a directed (a, b)-path $\overrightarrow{R}$ in A. Since the inserted segment of $\overrightarrow{C}$ has at most $|V(C)|-1$ edges, we have that $\ell (\overrightarrow{R}) \le \ell (R') + |V(C)|-1$, where $|V(C)| \le l+k$. We consider the following cases.

• (i) $a,b \in V(T)$.

  From (3) and the above considerations we get

  $$\begin{aligned} \begin{aligned} d_{A}(a,b)&\le {\left\{ \begin{array}{ll} \frac{k+1}{2}(p-l) -1 + |V(C)|-1&{} \quad \text {for }p-l\text { even,} \\ \frac{k+1}{2}(p-l-1) + |V(C)|-1 &{} \quad \text {for }p-l\text { odd,} \\ \end{array}\right. }\\&\le {\left\{ \begin{array}{ll} \frac{k+1}{2}(p-l) -1 + l+k-1&{} \quad \text {for }p-l\text { even,} \\ \frac{k+1}{2}(p-l-1) +l+k-1&{} \quad \text {for }p-l\text { odd,} \\ \end{array}\right. }\\&= {\left\{ \begin{array}{ll} \frac{k+1}{2}p -\frac{k-1}{2}l+k-2&{} \quad \text {for }p-l\text { even,} \\ \frac{k+1}{2}p -\frac{k-1}{2}l+\frac{k-3}{2} &{} \quad \text {for }p-l\text { odd.} \\ \end{array}\right. } \end{aligned} \end{aligned}$$

  If l is even, the above inequality becomes

  $$\begin{aligned} d_{A}(a,b) \le {\left\{ \begin{array}{ll} \frac{k+1}{2}p -\frac{k-1}{2}l+k-2&{} \quad \text {for }p\text { even,} \\ \frac{k+1}{2}p -\frac{k-1}{2}l+\frac{k-3}{2} &{} \quad \text {for }p\text { odd.} \\ \end{array}\right. } \end{aligned}$$

  Since $l \ge 2$, we obtain

  $$\begin{aligned} d_A(a,b) \le \left\{ \begin{array}{ll} \frac{k+1}{2}p -1 &{} \quad \text {if }p\text { is even,} \\ \frac{k+1}{2}p - \frac{k+1}{2} &{} \quad \text {if }p\text { is odd,} \end{array} \right. \end{aligned}$$

  and so (1) is satisfied. If l is odd then $l \ge 3$ and a similar calculation shows that again (1) holds.

• (ii) $a,b \in V(A)$.

  From (4) and the above considerations we get

  $$\begin{aligned} \begin{aligned} d_A(a,b)&\le {\left\{ \begin{array}{ll} \frac{k+1}{2}(p-l) +(l+k-1)+ k-2 &{} \quad \text {for }p-l\text { even,} \\ \frac{k+1}{2}(p-l) +(l+k-1)+ \frac{k-3}{2} &{} \quad \text {for }p-l\text { odd,} \\ \end{array}\right. } \\&= {\left\{ \begin{array}{ll} \frac{k+1}{2}p -(l-2)\frac{k-1}{2}+ k-2 &{} \quad \text {for }p-l\text { even,} \\ \frac{k+1}{2}p -(l-2)\frac{k-1}{2}+ \frac{k-3}{2} &{} \quad \text {for }p-l\text { odd.} \\ \end{array}\right. } \end{aligned} \end{aligned}$$

  When l is even, we have $l \ge 2$ and the above inequality becomes

  $$\begin{aligned} d_A(a,b) \le \left\{ \begin{array}{cc} \frac{k+1}{2}p +k-2 &{} \quad \text {if }p\text { is even,} \\ \frac{k+1}{2}p + \frac{k-3}{2} &{} \quad \text {if }p\text { is odd,} \end{array} \right. \end{aligned}$$

  and so (2) is clearly satisfied. If l is odd, then $l \ge 3$ and by a similar argument we show that (2) holds also in this case.

Case 2: Every T-path in ${{\mathcal {P}}}$ covers exactly one edge of T.

From here on we denote a T-path in ${{{\mathcal {P}}}}$ that covers an edge uv of T by $P_{uv}$. We also denote a T-cycle with end u by $P_u$. Note that an edge of T may have more than one T-path covering it, or a vertex of T may be in more than one T-cycle.

Case 2A: There exist disjoint edges $uv, wx \in E(T)$ for which ${{{\mathcal {P}}}}$ contains T-paths $P_{uv}$ and $P_{xw}$ that share a vertex.

Assume that u and x are the vertices in $\{u,v,w,x\} \subseteq V(T)$ that are furthest apart in T. Let $P_{uv}$ be the T-path $p_0, \ldots , p_r$ and $P_{xw}$ the T-path $q_0, \ldots , q_s$, where $1\le r,s \le k$, where $p_0=u$, $p_r=v$, $q_0=x$ and $q_s=w$. By the defining condition of Case 2A, $P_{uv}$ and $P_{xw}$ share a vertex, i.e., $p_i=q_j$ for some $0\le i \le r$ and $0\le j \le s$. Without loss of generality, let $p_i$ be such a vertex with the smallest subscript. Then $i \le j$ and let $Q_{uw}$ be the walk $p_0, p_1, \ldots , p_i, q_{j+1}, q_{j+2}, \ldots , q_s$. Since $p_0, \ldots , p_{i-1}$ are not on $P_{xw}$, $Q_{uw}$ is a path. $Q_{uw}$ has length $i+s-j$ and since $s\le k$ and $i \le j$, this is no greater than k. Hence, $Q_{uw}$ is a T-path that covers uv as well as any other edges between v and w. To ${{\mathcal {P}}}$ we add the T-path $Q_{uw}$. Note that this does not change the graph H. Since $Q_{uw}$ covers more than one edge of T, we can apply Case 1, and the lemma follows.

Case 2B: There exist distinct nonadjacent vertices $u,v \in V(T)$ for which ${{{\mathcal {P}}}}$ contains T-cycles $P_{u}$ and $P_{v}$ that share a vertex.

Assume that $P_u$ and $P_v$ share a vertex x. Then $P_u$ contains a (u, x)-path of length at most k/2, and $P_v$ contains an (x, v)-path of length at most k/2. Combining these two paths we obtain a (u, v)-path $P_{uv}$ of length at most k. Since u and v are nonadjacent, $P_{uv}$ covers more than one edge. As in Case 2A we add $P_{uv}$ to ${{{\mathcal {P}}}}$ without changing H, and we apply Case 1.

Case 2C: Whenever T-paths $P_{e_1}, P_{e_2} \in {{{\mathcal {P}}}}$ share a vertex, then $e_1$ and $e_2$ are not disjoint, and whenever T-cycles $P_u, P_v \in {{{\mathcal {P}}}}$ share a vertex, then u and v are equal or adjacent.

We first modify ${{{\mathcal {P}}}}$ so that for every edge e of T there exists a path in ${{{\mathcal {P}}}}$ covering e. If $uv \in E(T)$ is an edge that is not covered, then the defining condition of Case 2C implies that ${{{\mathcal {P}}}}$ contains T-cycles $P_u$ and $P_v$ that share a vertex, otherwise e would be a bridge of H. Let x be a common vertex of $P_u$ and $P_v$. Then $P_u$ contains a (u, x)-path of length at most k/2 and $P_v$ contains an (x, v)-path of length at most k/2. Combining these two paths yields a path $P_{uv}$ of length at most k. Adding this path to ${{{\mathcal {P}}}}$ does not change H. Adding such a path to ${{{\mathcal {P}}}}$ for each edge of T that is not covered yields a k-extension of T in which every edge of T is covered by a T-path.

Now T is a tree and thus bipartite; we denote by U and W the partite sets of T. Let $F = \bigcup _{P \in {{{\mathcal {P}}}}}P$ and for an edge uw of T let $R_{uw}$ be a shortest (u, w)-path among all (u, w)-paths in F that have no internal vertex in T. Let ${{{\mathcal {R}}}}$ be the set of all such shortest paths. Clearly, ${{{\mathcal {R}}}}$ is a k-extension of T and each T-path in ${{{\mathcal {R}}}}$ covers exactly one edge of T. If ${{{\mathcal {R}}}}$ contains two paths that cover disjoint edges of T and share a vertex, then we can apply Case 2A to ${{{\mathcal {R}}}}$. Hence, we may assume that the defining condition of Case 2C applies, i.e., $R_{uv}$ and $R_{wx}$ are vertex-disjoint whenever uv and wx are two disjoint edges of T. Let $R = \bigcup _{uw \in E(T)}R_{uw}$. We now define an orientation A of the edges of $R\cup T$. Let $uw \in E(T)$, where $u\in U$ and $w \in W$. Orient the edge uw as $\overrightarrow{uw}$. Orient the edges of $R_{uw}$ from w to u, that is, if ab is an edge of $R_{uw}$ with a being closer to u and b being closer to w, then orient ab as $\overrightarrow{ba}$. We denote the resulting directed paths by $\overrightarrow{R}_{uw}$ and $\overrightarrow{R}_{wu}$, respectively. We now show that no edge of R receives conflicting orientations. This clearly holds for the edges of T. Suppose to the contrary that there exists an edge ab that lies on $R_{uw}$ and $R_{u'w'}$, where $u,u' \in U$ and $w, w' \in W$, that receives conflicting orientations. We may assume that on $R_{uw}$ vertex a is closer to u and b is closer to w, while on $R_{u'w'}$ vertex b is closer to u and a is closer to w. By the defining condition of Case 2C, either $u'=u$ or $w'=w$. We may assume the former. $R_{uw}$ is a shortest (u, w)-path in R and the (u, a)-path is shorter than the (u, b)-path since a is closer to u than b, so $d_R(u,a)<d_R(u,b)$. However, $R_{uw'}$ is a shortest $(u,w')$-path in R and the (u, a)-path is longer than the (u, b)-path since b is closer to u than a, so $d_R(u,a) > d_R(u,b)$. This contradiction shows that no edge of R receives two conflicting orientations.

We now bound the distances between any two vertices of T in A. Let $Q:v_0, v_1, \ldots , v_d$ be a $(v_0, v_d)$-path in T. Q can be turned into a directed $(v_0,v_d)$-path $\overrightarrow{Q}$ in A by replacing the edge $v_iv_{i+1}$ by the directed path $\overrightarrow{R}_{v_iv_{i+1}}$. If d is odd then $v_0$ and $v_d$ must be in different partite sets. When $v_0 \in U$ and $v_d \in W$, $\overrightarrow{Q}$ consists of exactly $(d+1)/2$ directed edges and $(d-1)/2$ directed paths of length at most k. Hence, $\ell (\overrightarrow{Q}) \le \frac{d+1}{2} + \frac{d-1}{2}k$. On the other hand, when $v_0 \in W$ and $v_d \in U$, then by a similar argument $\ell (\overrightarrow{Q}) \le \frac{d-1}{2} + \frac{d+1}{2}k$. When d is even, $v_0$ and $v_d$ are in the same partite sets. Then $\overrightarrow{Q}$ has exactly d/2 directed edges and d/2 directed paths of length at most k and so $\ell (\overrightarrow{Q}) \le \frac{d}{2} + \frac{d}{2}k$. Thus, for any $x, y \in V(T)$ with $d_T(x,y)=d$ we get

$$\begin{aligned} d_A(x,y) \le {\left\{ \begin{array}{ll} \frac{d}{2}(k+1) &{}\quad \text {if }x,y \in U\text { or }x,y \in W, \\ \frac{d+1}{2} + \frac{d-1}{2}k &{}\quad \text {if }x \in U\text { and }y \in W, \\ \frac{d-1}{2} + \frac{d+1}{2}k &{}\quad \text {if }x \in W\text { and }y \in U. \end{array}\right. } \end{aligned}$$

Hence,

$$\begin{aligned} d_A(x,y) \le \bigg \lfloor \frac{d}{2} \bigg \rfloor + \bigg \lceil \frac{d}{2} \bigg \rceil k. \end{aligned}$$

(5)

Substituting $d \le p-1$ and simplifying, we get

$$\begin{aligned} d_A(x,y) \le \left\{ \begin{array}{ll} \frac{k+1}{2}p-1 &{} \quad \text {for }p\text { even,} \\[2pt] \frac{k+1}{2}(p - 1) &{} \quad \text {for }p\text { odd,} \end{array} \right. \end{aligned}$$

and so (1) is satisfied.

We now bound the distances between any two vertices $x, y \in V(A)$.

• (i) $x \in V(T), y \notin V(T)$.

  Vertex y is on the path $R_{v_iv_j}$ for some $v_i, v_j \in V(T)$. Let $v_i$ be closer to x in T than $v_j$, and let $d:=d_T(x,v_i)$. Then $d_A(y,v_i) \le k$ since y and $v_i$ are on the directed cycle $\overrightarrow{R}_{yv_i} \cup \overrightarrow{R}_{v_i y}$ which has length at most $k+1$. It follows from (5) that

  $$\begin{aligned} d_A(x,y) \le d_A(x,v_i) + d_A(v_i, y) \le \bigg \lfloor \frac{d}{2} \bigg \rfloor + \bigg \lceil \frac{d}{2} \bigg \rceil k+k. \end{aligned}$$

  Now $d \le p-2$. Substituting this value yields, after simplification, that

  $$\begin{aligned} d_A(x,y) \le \left\{ \begin{array}{ll} \frac{k+1}{2}p-1 &{} \quad \text {for }p\text { even,} \\ \frac{k+1}{2}p + \frac{k-3}{2} &{} \quad \text {for }p\text { odd,} \end{array} \right. \end{aligned}$$

  which implies (2).

• (ii) $x \notin V(T), y \in V(T)$.

  The same proof as in i, with the roles of x and y exchanged, yields (2).

• (iii) $x,y \notin V(T)$.

  Vertex x is on the path $R_{v_iv_j}$ and y is on the path $R_{v_sv_t}$ for some $v_i, v_j, v_s, v_t \in V(T)$. Let $v_s$ be closer to x in T than $v_t$ and $v_i$ closer to y in T than $v_j$ and let $d:=d_T(v_i,v_s)$. Then $d_A(x,v_i) \le k$ since x and $v_i$ are on the directed cycle $\overrightarrow{R}_{xv_i} \cup \overrightarrow{R}_{v_i x}$ which has length at most $k+1$. Similarly, $d_A(y,v_s) \le k$. It follows from (5) that

  $$\begin{aligned} d_A(x,y) \le d_A(x,v_i) +d_A(v_i, v_s)+ d_A(v_s, y) \le k+\bigg \lfloor \frac{d}{2} \bigg \rfloor + \bigg \lceil \frac{d}{2} \bigg \rceil k+k. \end{aligned}$$

  Here $d \le p-3$, so by a similar argument as in Case i, we get that

  $$\begin{aligned} d_A(x,y) \le \left\{ \begin{array}{cc} \frac{k+1}{2}p+k-2 &{} \quad \text {for }p\text { even,} \\ \frac{k+1}{2}p + \frac{k-3}{2} &{} \quad \text {for }p\text { odd,} \end{array} \right. \end{aligned}$$

  and (2) holds for all 3 cases. Having shown that the bounds are satisfied for Case 2, the proof of the lemma is complete. $\Box$

4 Oriented Diameter in Terms of Connected Domination Number

We now use Lemma 1 to yield the first main result of this paper. We split the proof of our main result into two parts. We first prove the upper bound on the oriented diameter, and then we construct graphs that show this bound is sharp.

We make use of the following result by Dankelmann et al. [8].

Theorem 1

(Dankelmann, Guo and Surmacs [8]) Let G be a bridgeless graph of order n and maximum degree $\varDelta$. Then

$$\begin{aligned} \overrightarrow{{\textrm{diam}}}(G) \le n - \varDelta +3. \end{aligned}$$

Theorem 2

For every bridgeless graph G,

$$\begin{aligned} \overrightarrow{{\textrm{diam}}}(G) \le {\left\{ \begin{array}{ll} 2 \gamma _c(G) +3 &{} \quad \text {for }\gamma _c(G)\text { even,} \\ 2 \gamma _c(G) +2 &{} \quad \text {for }\gamma _c(G)\text { odd}. \\ \end{array}\right. } \end{aligned}$$

(6)

Proof

Let G be a bridgeless graph and S a connected dominating set with $\vert S \vert = \gamma _c(G)$. If $\gamma _c(G)=1$, then G contains a vertex of degree $n-1$, where n is the order of G. It follows from Theorem 1 that $\overrightarrow{{\textrm{diam}}}(G) \le 4$, and so the theorem holds in this case. Hence, we assume that $\gamma _c (G)\ge 2$.

Let T be a spanning tree of G[S]. Each vertex v not in T has a neighbour $v'$ in T. We extend T to a spanning tree $T_1$ of G by adding each $v \in V(G) - V(T)$ and the edge $vv'$. If $uv \in E(T)$, then $uv \in E(T_1)$. Let $T^u$ and $T^v$ be the components in $T - \{uv\}$ containing u and v, respectively. The graph $T_1 - \{uv\}$ consists of two components, $T_1^u$ containing u and $T_1^v$ containing v. Since G is bridgeless, $T_1^u$ and $T_1^v$ are connected by an edge $u'v' \in E(G) - E(T_1)$. Then $T_1-uv+u'v'$ contains a (u, v)-path. On this (u, v)-path, let $p_u$ be the last vertex in $T^u$ and $p_v$ the first vertex in $T^v$. Call the segment between $p_u$ and $p_v$, that is not in T, $P_{uv}$. Since S is a dominating set, $P_{uv}$ has no internal vertices other than possibly $u'$ and $v'$. Every edge of T is covered and so the graph $T \cup \bigcup _{e\in E(T)} P_e$ is bridgeless. Since $P_{uv}$ is a T-path of length at most 3, ${{{\mathcal {P}}}}$ is a 3-extension of T.

Let $H = T \cup \bigcup _{P \in {{{\mathcal {P}}}}} P$. Applying Lemma 1, with $k=3$ and $p= \gamma _c(G)$, there exists a strong orientation $A'$ of a subgraph of H containing T such that

• (i) for every two vertices $u,v \in V(T)$,

  $$\begin{aligned} d_{A'}(u,v) \le {\left\{ \begin{array}{ll} 2\gamma _c(G) -1 &{} \quad \text {for }\gamma _c(G)\text { even,} \\ 2(\gamma _c(G)-1) &{} \quad \text {for }\gamma _c(G)\text { odd,} \\ \end{array}\right. } \end{aligned}$$

  (7)

• (ii) for every two vertices $u,v \in V(A')$,

  $$\begin{aligned} d_{A'}(u,v) \le {\left\{ \begin{array}{ll} 2\gamma _c(G) +1 &{} \quad \text {for }\gamma _c(G)\text { even,} \\ 2\gamma _c(G) &{} \quad \text {for }\gamma _c(G)\text { odd.} \\ \end{array}\right. } \end{aligned}$$

  (8)

We now extend $A'$ to a strong orientation A of G such that

$$\begin{aligned} \begin{aligned}&\text {every vertex not in }A\text { is at distance at most 2 both to and from}\\&\text {the nearest vertex of }T. \end{aligned} \end{aligned}$$

(9)

Let U be the set of all vertices that are not in A and that have at least two neighbours in T. For every $u\in U$ choose a neighbour $u' \in V(T)$ and orient the edge $uu'$ as $\overrightarrow{uu'}$, and for all other neighbours v of u in V(T) orient the edge uv as $\overrightarrow{vu}$. Let Q be a connected component of $G-(V(A') \cup U)$. Clearly, Q consists of exactly one vertex or at least two vertices.

If Q consists of one vertex u, then u is adjacent to exactly one vertex in T, say $y \in V(T)$. Since G is bridgeless u is adjacent to a vertex $w \in V(A') \cup U$. Now w is adjacent to a vertex $x \in V(T)$ and since $w \in V(A') \cup U$ the edge wx may have already received an orientation. If wx has been oriented as $\overrightarrow{wx}$ then we orient the edges uy and uw as $\overrightarrow{yu}$ and $\overrightarrow{uw}$, respectively. Conversely, if wx has been oriented as $\overrightarrow{xw}$ then we orient the edges uy and uw as $\overrightarrow{uy}$ and $\overrightarrow{wu}$. If wx has not received an orientation then we may orient wx as $\overrightarrow{wx}$ or $\overrightarrow{xw}$, ensuring that the edges uy and uw are oriented accordingly, as described above. Clearly, u is distance at most 2 both to and from a vertex in T and so (9) is satisfied.

Now suppose that there are at least two vertices in the connected component Q. Let $T_Q$ be a spanning tree of Q and let $U_Q$ and $W_Q$ be its partite sets. Orient the edges of $T_Q$ from $U_Q$ to $W_Q$. For each vertex $v \in V(T_Q)$, v has exactly one neighbour $v'$ in T. If v is in $U_Q$ then orient the edge $vv'$ as $\overrightarrow{v'v}$ and if v is in $W_Q$, then orient the edge $vv'$ as $\overrightarrow{vv'}$. Now each vertex in Q is distance at most 2 both to and from a vertex in T and again (9) is satisfied.

We are now in a position to apply inequalities (7) and (8) to derive (6). Assume that $\gamma _c(G)$ even. We consider the following different cases for the distances between every pair of vertices $u, w \in V(G)$ in A.

Case 1: $u,w \in V(A)$.

From (8) we get that $d_A(u,w) \le 2\gamma _c(G)+1$, as desired.

Case 2: $u,w \in V(G) - V(A)$.

Let a and b be vertices of T that are at distance at most 2 from u and to w, respectively. Then by (7), $d_A(u,w) \le d_A(u,a) + d_A(a,b) + d_A(b,w) \le 2 + 2\gamma _c(G) -1 + 2 = 2\gamma _c(G) +3$, as desired.

Case 3: $u \in V(A)$ and $w \in V(G)-V(A)$.

Let $a \in V(T)$ such that a is at distance at most 2 from $w \notin V(T)$. By (9), $d_A(w,a) \le 2$. We can then apply (8) and bound the distance between u and w by $d_A(u,w) \le d_A(u,a)+d(a,w) \le 2\gamma _c(G)+1+2 = 2\gamma _c (G)+3$, as desired.

Case 4: $w \in V(A)$ and $u \in V(G)-V(A)$.

Exchanging the roles of u and w leads to the previous case and $d_A(u,w) \le 2\gamma _c(G) +3$.

Hence, inequality (6) follows in all cases if $\gamma _c(G)$ is even. We omit the proof for the case that $\gamma _c(G)$ is odd as it is very similar. The theorem follows. $\square$

Theorem 2 is sharp. To show this we construct, for given $p \in {\mathbb {N}}$, a graph $G_p$ with $\gamma _c(G_p)=p$ that attains the bound in Theorem 2.

We construct $G_p$ as follows. For $p=1$, $G_1$ consists of a single vertex $a_1$ with two edge-disjoint triangles attached to $a_1$. Clearly, $\vert V(G_1) \vert = 5$ and $\overrightarrow{{\textrm{diam}}}(G_1)=4$ and so $G_1$ attains the bound in (6). For $p \ge 2$, $p \in {\mathbb {N}}$, let $T_p$ be the path $a_1, a_2, \ldots , a_p$. For each edge $a_i a_{i+1}$, $1 \le i < p$, attach a path $P_{a_ia_{i+1}}$ of length 3 whose initial vertex is $a_i$ and whose terminal vertex is to $a_{i+1}$, thus forming a 4-cycle for each edge of $T_p$. Append a triangle $Q_1$ to $a_1$, rooted at $a_1$ and similarly, $Q_p$ to $a_p$. Define $G_p$ to be the graph $G_p=T_p\cup \bigcup _{i=1}^{p-1} P_{a_ia_{i+1}} \cup Q_1 \cup Q_p$. Figures 1 and 2 show $G_p$ with an orientation of $G_6$ and $G_5$, respectively. The vertices of $T_p$ are solid, the internal vertices of $P_{a_ia_{i+1}}$ are white and the grey vertices are those in $Q_1$ and $Q_p$.

Fig. 1

A strong orientation of the graph $G_6$

Fig. 2

A strong orientation of the graph $G_5$

Proposition 1

Let $G_p$ be the graph constructed above. Then

$$\begin{aligned} \overrightarrow{{\textrm{diam}}}(G_p) = {\left\{ \begin{array}{ll} 2 \gamma _c(G_p) +3 &{} \quad \text {for }\gamma _c(G_p)\text { even,} \\ 2 \gamma _c (G_p)+2 &{} \quad \text {for }\gamma _c(G_p)\text { odd}. \\ \end{array}\right. } \end{aligned}$$

Proof

We first note that $\gamma _c(G_p)=p$. Indeed, every connected dominating set must contain all cut-vertices of $G_p$, and thus every vertex of $T_p$. On the other hand, the vertices of $T_p$ form a connected dominating set. Hence $V(T_p)$ is a minimum connected dominating set of $G_p$, and so $\gamma _c(G_p)=|V(T_p)| =p$.

We now determine $\overrightarrow{{\textrm{diam}}}(G_p)$. Let A be an arbitrary strong orientation of $G_p$. Then a path from $a_1$ to $a_p$ must pass through the vertices $a_1, a_2, \ldots , a_p$ in that order, since each $a_i, 1 \le i \le p$, is a cut-vertex of $G_p$. Similarly, an $(a_p,a_1)$-path must pass through the vertices $a_p, a_{p-1}, \ldots , a_1$ in that order. Hence,

$$\begin{aligned} d_A(a_1, a_p) = d_A(a_1, a_2) + d_A(a_2, a_3) + \cdots + d_A(a_{p-1}, a_p) \end{aligned}$$

(10)

and

$$\begin{aligned} d_A(a_p, a_1) = d_A(a_p, a_{p-1}) + d_A(a_{p-1}, a_{p-2}) + \cdots + d_A(a_2, a_1). \end{aligned}$$

(11)

Since each edge of the 4-cycle through $a_ia_{i+1}$ is either on the $(a_i, a_{i+1})$-path or on the $(a_{i+1}, a_i)$-path in A, we have that $d_A(a_i, a_{i+1})+d_A(a_{i+1}, a_i)=4$. Hence, adding together (10) and (11) we get that

$$\begin{aligned} d_A(a_1,a_p)+d_A(a_p,a_1)=4(p-1). \end{aligned}$$

(12)

From (12) we get that

$$\begin{aligned} \max \{d_A(a_1,a_p),d_A(a_p,a_1)\} \ge 2(p-1)=2p-2. \end{aligned}$$

Now for each $a_i, 1 \le i < p$, we have that $d_A(a_i, a_{i+1})$ and $d_A(a_{i+1}, a_i)$ are odd. We assume that p is even and give the details only for this case. A similar argument holds for p odd. If p is even, then we have an odd number of distances from each $a_i$ to $a_{i+1}$ and so $d_A(a_1, a_p)$, and similarly, $d_A(a_p, a_1)$, is odd. Hence, $\max \{d_A(a_1,a_p),d_A(a_p,a_1)\} \ge 2p-1$. We may assume that $d_A(a_p, a_1) \ge d_A(a_1, a_p)$. Then

$$\begin{aligned} d_A(a_p, a_1) \ge 2p-1. \end{aligned}$$

(13)

Since A is strong $Q_1$ and $Q_p$ are oriented as directed 3-cycles. Let $q_1$ be the in-neighbour of $a_1$ in $V(Q_1)$ and $q_p$ the out-neighbour of $a_p$ in $V(Q_p)$. Then $d_A(a_1, q_1)=2$ and $d_A(q_p, a_p)=2$, so by (13)

$$\begin{aligned} \overrightarrow{{\textrm{diam}}}(G_p) \ge d_A(q_p,q_1) = d_A(q_p,a_p) + d_A(a_p, a_1) + d_A(a_1, q_1) \ge 2p+3. \end{aligned}$$

Since $\overrightarrow{{\textrm{diam}}}(G_p) \le 2p+3$ by Theorem 2, we have $\overrightarrow{{\textrm{diam}}}(G_p) =2p+3$, as desired. $\square$

It follows from Proposition 1 that the bound in Theorem 2 is sharp.

5 A generalisation of connected domination

In this section we generalise the ideas of the previous section to obtain an upper bound on the oriented diameter in terms of a new parameter, the connected d-distance domination number. This result will be used in the final section to obtain a bound on the oriented diameter in terms of the d-distance domination number. See the relevant definitions below.

Def 3

For a given integer $d\in {\mathbb {N}}$, a d-distance dominating set S of a graph G is a set of vertices such that every vertex of G is within distance d of some vertex of S.

The d-distance domination number $\gamma ^d(G)$ of a graph G is the minimum cardinality of a d-distance dominating set.

Def 4

For a given integer $d\in {\mathbb {N}}$, a connected d-distance dominating set S of a graph G is a d-distance dominating set that induces a connected subgraph in G.

The connected d-distance domination number $\gamma _c^d(G)$ of a graph G is the minimum cardinality of a connected d-distance dominating set.

Clearly, $\gamma _c(G) = \gamma _c^1(G)$ for every graph G, so the d-distance connected domination number generalises the connected domination number.

In the proof of the main result of this section, Theorem 4, we make use of Lemma 1. In addition, we make use of the following result due to Chvátal and Thomassen, contained in Theorem 2 in [4].

Theorem 3

(Chvátal and Thomassen [4]) If G is a bridgeless multigraph and if u is a vertex of G such that $d_G(u,v) \le r$ for every vertex v, then there is an orientation A of G such that $d_A(u,v) \le r^2+r$ and $d_A(v,u) \le r^2+r$ for every vertex v.

Theorem 4

For every bridgeless graph G with connected d-distance domination number $\gamma _c^d(G)$,

$$\begin{aligned} \overrightarrow{{\textrm{diam}}}(G) \le {\left\{ \begin{array}{ll} \gamma _c^d(G)(d+1) +2d^2+4d-1 &{}\quad \text {for }\gamma _c^d(G)\text { even,} \\ \gamma _c^d(G)(d+1) +2d^2+3d-1 &{} \quad \text {for }\gamma _c^d(G)\text { odd}. \\ \end{array}\right. } \end{aligned}$$

Proof

Let G be a bridgeless graph and S a connected d-distance dominating set with $\vert S \vert = \gamma _c^d(G)$. If $\gamma _c^d(G)=1$, then the radius of G is at most d. By Theorem 3 we get that $\overrightarrow{{\textrm{rad}}}(G) \le d^2+d$, which in turn implies that $\overrightarrow{{\textrm{diam}}}(G) \le 2(d^2+d) < 2d^2+4d$. Therefore, the bound of Theorem 4 holds when $\gamma _c^d(G)=1$. Thus, we consider the condition when $\gamma _c^d(G) \ge 2$.

Let T be a spanning tree of G[S]. We construct an orientation $A'$ of a subgraph of G containing T in almost exactly the same way as in Theorem 2. In this case we extend T to a spanning tree $T_1$ of G that preserves the distances to the vertices of T. Making use of $T_1$, we then get a $(2d+1)$-extension ${{{\mathcal {P}}}}$ of T. Applying Lemma 1, we obtain a strong orientation $A'$ of a subgraph containing T such that for every two vertices $u,v \in V(A')$,

$$\begin{aligned} d_{A'}(u,v) \le {\left\{ \begin{array}{ll} \gamma _c^d(G)(d+1) + 2d-1 &{} \quad \text {for }\gamma _c^d(G)\text { even,} \\ \gamma _c^d(G)(d+1) +d-1 &{} \quad \text {for }\gamma _c^d(G)\text { odd}. \\ \end{array}\right. } \end{aligned}$$

(14)

In G, we identify all the vertices of $V(A')$ to a single vertex $u^*$. This may create multiple edges; however, we delete any loops that arise and call this new multigraph $G^*$. Note that $G^*$ is bridgeless and every vertex in $G^*$ is within distance d of $u^*$. By Theorem 3 there exists an orientation $A^*$ in which $u^*$ has eccentricity at most $d^2+d$. We obtain the orientation A of G by combining $A'$ and $A^*$ as follows. The edges oriented in $A'$ retain their orientations in G. Each of the remaining edges of G corresponds to some edge of $G^*$, and it receives the same orientation as the corresponding edge in $A^*$. Now, every vertex not in $A'$ is within distance at most $d^2+d$ both to and from some vertex in $A'$. Hence, from (14), we obtain that

$$\begin{aligned} \begin{aligned} \overrightarrow{{\textrm{diam}}}(A)&\le \overrightarrow{{\textrm{diam}}}(A')+2(d^2+d)\\&\le {\left\{ \begin{array}{ll} \gamma _c^d(G)(d+1) + 2d-1 +2(d^2+d) &{} \quad \text {for }\gamma _c^d(G)\text { even,} \\ \gamma _c^d(G)(d+1) +d-1 +2(d^2+d)&{} \quad \text {for }\gamma _c^d(G)\text { odd}, \\ \end{array}\right. }\\&= {\left\{ \begin{array}{ll} \gamma _c^d(G)(d+1) + 2d^2 + 4d-1 &{} \quad \text {for }\gamma _c^d(G)\text { even,} \\ \gamma _c^d(G)(d+1) +2d^2+ 3d-1 &{} \quad \text {for }\gamma _c^d(G)\text { odd}, \\ \end{array}\right. } \end{aligned} \end{aligned}$$

as desired. $\square$

We now present a graph whose oriented diameter differs from the bound in Theorem 4 by 2d. Our construction is similar to that of the graph $G_p$ in the previous section, but in addition it relies on a graph constructed by Chvátal and Thomassen [4] to show that Theorem 4 above is sharp, apart from an additive constant that depends only on d.

Lemma 2

(Chvátal and Thomassen [4]) Given $r \in {\mathbb {N}}$, there exists a bridgeless graph $H_r$ and a vertex $u_r$ of $H_r$ such that ${\textrm{ecc}}_{H_r}(u_r) =r$, and ${\textrm{ecc}}_{\overrightarrow{H_r} }(u_r)\ge r^2+r$ in every strong orientation $\overrightarrow{H_r}$.

For completeness we briefly describe the construction of the graph $H_r$. We give a slightly simplified version of the graph that is of interest to us. A sequence of rooted graphs, $H_1, H_2,\ldots$ is constructed as follows. For radius r, $H_r$ can be constructed by taking a cycle $u_0, u_1, \ldots , u_{2r}, u_0$ and appending two disjoint copies of $H_{r-1}$ by identifying the root of the first copy to $u_1$ and the second copy to $u_{2r}$. $H_r$ is rooted at $u_0$. A strong orientation of the graph $H_3$ is shown in Fig. 3. Figure 3 shows the vertices $u_1$ and $u_6$ to which copies of $H_2$ are attached. It is easy to see that every strong orientation $\overrightarrow{H_r}$ is obtained by orienting every cycle as a directed cycle, and that all strong orientations of $H_r$ are isomorphic.

Fig. 3

A strong orientation of the graph $H_3$

We now construct the graph $G_p^d$ for given $d,p \in {\mathbb {N}}$, $p \ge 2$, as follows. $T_p$ is the path $a_1,\ldots , a_p$ and for each edge $a_ia_{i+1}$ we add an $(a_i, a_{i+1})$-path $P_{a_i, a_{i+1}}$ of length $2d+1$. We identify both $a_1$ and $a_p$ with the root of a copy of the graph $H_d$ in Lemma 2. Figure 4 shows the graph $G_6^2$. Taking $S=\{a_1, a_2, \ldots a_p\}$, we conclude that $\gamma _c^d(G_p^d)=p$.

Let A be a strong orientation of $G_p^d$. We may assume that $d_A(a_p, a_1) \ge d_A(a_1, a_p)$. Thus, for at least $\lceil \frac{p-1}{2} \rceil$ values of i, where $1 \le i < p$, we have $d_A(a_{i+1}, a_i)=2d+1$ and for the remaining values of i we have $d_A(a_{i+1}, a_i) =1$, Hence,

$$\begin{aligned} d_A(a_p, a_1) \ge \left\lceil \llceil \frac{p-1}{2}\right\rceil \rrceil (2d+1) +\left\lfloor \frac{p-1}{2}\right\rfloor . \end{aligned}$$

By Lemma 2, $Q_1$ contains a vertex $q_1$ with $d_A(a_1,q_1)\ge d^2+d$ and $Q_p$ contains a vertex $q_p$ with $d_A(q_p,a_p)\ge d^2+d$, so

$$\begin{aligned} \begin{aligned} \overrightarrow{{\textrm{diam}}}(G_p^d)&\ge d_A(q_p,q_1) = d_A(q_p,a_p) + d_A(a_p, a_1) + d_A(a_1, q_1)\\&\ge \left\lceil \llceil \frac{p-1}{2}\right\rceil \rrceil (2d+1) +\left\lfloor \frac{p-1}{2}\right\rfloor + 2d^2+2d. \end{aligned} \end{aligned}$$

Since there exists an orientation of $G_p^d$ with this diameter, we get, with $p=\gamma _c^d(G_p^d)$ that

and we conjecture that the value of is in fact a sharp upper bound.

Fig. 4

The graph $G_6^2$

6 Oriented diameter in terms of distance domination number

In this section we apply Theorem 4 to obtain a bound on the oriented diameter of a bridgeless graph in terms of the d-distance domination number. We show that for every $d \in {\mathbb {N}}$ the oriented diameter is bounded from above by a term of the form $a_d \gamma ^d(G) +c_d$, where $a_d$ and $c_d$ are constants that depend only on d and not on $\gamma ^d(G)$. We determine the best possible value of $a_d$ within a factor of about two.

Corollary 1

Let G be a bridgeless graph of order n and $d \in {\mathbb {N}}$. Then

$$\begin{aligned} \overrightarrow{{\textrm{diam}}}(G) \le \left\{ \begin{array}{ll} (2d+1)(d+1)\gamma ^d(G) + 2d-1 &{} \quad \text {if }\gamma ^d(G)\text { is even}, \\ (2d+1)(d+1)\gamma ^d(G) + d-1 &{} \quad \text {if }\gamma ^d(G)\text { is odd}. \end{array} \right. \end{aligned}$$

Proof

We first show that

$$\begin{aligned} \gamma _c^d(G) \le (2d+1)\gamma ^d(G) - 2d. \end{aligned}$$

(15)

Let $p:= \gamma _c^d(G)$ and let S be a d-distance dominating set of G of cardinality p. Choose a vertex $v_1 \in S$ and let $U_1 = \{v_1\}$. If $p\ge 2$, then there exists a vertex $v_2 \in S$ at distance at most $2d+1$ from $v_1$ since otherwise no vertex at distance $d+1$ from $v_1$ would have a vertex in S within distance d. Let $P_2$ be a shortest $(v_2,v_1)$-path in G and let $U_2=U_1\cup V(P_2)$. Now there exists a vertex $v_3 \in S$ at distance at most $2d+1$ from some $v_i \in \{v_1,v_2\}$ since otherwise no vertex at distance $d+1$ from $\{v_1,v_2\}$ would have a vertex in S within distance d. Let $P_3$ be a shortest path from $v_3$ to this $v_i$ in G and let $U_3=U_2 \cup V(P_3)$. Repeating this argument, we eventually obtain a set $U_p$ containing S. Clearly, $U_p$ is a d-distance dominating set and induces a connected subgraph in G. Moreover, since $|U_1|=1$ and each $P_i$ adds at most $2d+1$ new vertices to $U_{i-1}$, we have $|U_p| \le (2d+1)p - 2d$, and (15) follows.

Now applying Theorem 4 yields the corollary. $\square$

The bound in Corollary 1 is likely not sharp. For given $d \in {\mathbb {N}}$ let $a_d$ be the smallest real number for which there exists a constant $c_d$ depending only on d such that for every bridgeless graph G,

$$\begin{aligned} \overrightarrow{{\textrm{diam}}}(G) \le a_d \gamma ^d(G) + c_d. \end{aligned}$$

Corollary 1 shows that $a_d \le (2d+1)(d+1)$. It appears difficult to determine the value of $a_d$. Even for $d=1$ the exact value of $a_1$ is not known. From the results mentioned in the introduction we know that $\frac{7}{2} \le a_1 \le 4$. The following proposition gives a lower bound for $a_d$.

Proposition 2

Let $d,p \in {\mathbb {N}}$. Then there exists a bridgeless graph $G^{d,p}$ which satisfies $\gamma ^d(G^{d,p})=p$ and

$$\begin{aligned} \overrightarrow{{\textrm{diam}}}(G^{d,p}) \ge p(d^2+2d+\frac{1}{2})-1. \end{aligned}$$

Proof

Let d and p be fixed. Let $P_d^*$ be the graph obtained from the disjoint union of a path of order $d+1$ with vertices $a_0, a_1, \ldots , a_d$ and paths of order $1,3,\ldots ,2d-1$ by joining the two ends of the path of order $2i+1$ to $a_i$ and $a_{i+1}, 0 \le i < d$, respectively, by an edge. We denote by $b_0$ the neighbour of $a_0$ that has degree 2. Let $P_d^{**}$ be obtained from two disjoint copies of $P_d^*$ by identifying the two vertices corresponding to $a_d$ to a new vertex u. We denote the two vertices corresponding to $a_0$ and $b_0$, respectively, in one copy of $P_d^*$ by v and w, and the vertex corresponding to $a_0$ in the other copy of $P_d^*$ by x.

Let $F_1, F_2,\ldots , F_p$ be disjoint copies of $P_d^{**}$ and denote the vertices of $F_i$ corresponding to u, v, w and x by $u_i, v_i, w_i$ and $x_i$, respectively. We construct $G^{d,p}$ from the union $F_1 \cup F_2 \cup \ldots F_p$ by adding the edges $x_iv_{i+1}$ and $x_i w_{i+1}$ for $i=1,2,\ldots ,p-1$. A strong orientation of the graph $G^{3,2}$ is shown in Fig. 5.

It is easy to see that the set $\{u_1, u_2,\ldots ,u_p\}$ is a minimal d-distance dominating set of $G^{d,p}$, so $\gamma ^d(G^{d,p})=p$. To determine the oriented diameter of $G^{d,p}$, note that in every strong orientation $\overrightarrow{P_d^*}$ of $P_d^*$, the $(a_d, a_0)$-path and the $(a_0, a_d)$-path are unique, and that their union contains all the edges of $\overrightarrow{P_d^*}$, so $d_{\overrightarrow{P_d^*}}(a_0,a_d) + d_{\overrightarrow{P_d^*}}(a_d,a_0) = |E(P_d^*)| = d^2+2d$. Similarly, the two paths between $a_d$ and $\{a_0,b_0\}$ contain every edge of $\overrightarrow{P_d^*}$ except possibly the directed edge arising from $a_0b_0$, hence $d_{\overrightarrow{P_d^*}}(\{a_0, b_0\},a_d) + d_{\overrightarrow{P_d^*}}(a_d,\{a_0, b_0\}) \ge |E(P_d^*)| - 1 = d^2+2d-1$. Applying this to $G^{d,p}$, we obtain that for every strong orientation A of $G^{d,p}$,

$$\begin{aligned} 2\, {\textrm{diam}}(A)\ge & {} d_A(v_1, x_p) + d_A(x_p,v_1) \\\ge & {} \sum _{i=1}^p\big ( d_A(\{v_i, w_i\}, u_i) + d_A(u_i, x_i) + d_A(x_i, u_i) + d_A(u_i, \{v_i, w_i\}) \big ) \\{} & {} + 2(p-1) \\\ge & {} \sum _{i=1}^p (2d^2 + 4d-1) + 2(p-1) \\= & {} p(2d^2+4d+1) - 1. \end{aligned}$$

Since A is an arbitrary strong orientation of $G^{d,p}$, and since $\gamma ^d(G^{d,p})=p$, we conclude that

$$\begin{aligned} {\overrightarrow{\textrm{diam}}}(G^{d,p}) \ge \gamma ^d(G^{d,p}) \left( d^2+2d+ \frac{1}{2}\right) -1, \end{aligned}$$

as desired. $\square$

Fig. 5

A strong orientation of the graph $G^{3,2}$ as constructed in Proposition 2

From Corollary 1 and Proposition 2 we conclude that for every $d\in {\mathbb {N}}$,

$$\begin{aligned} d^2 + 2d + +\frac{1}{2} \le a_d \le 2d^2 + 3d+1, \end{aligned}$$

which determines $a_d$ within a factor of about 2. While it seems to be difficult to determine the value of $a_d$ precisely, it may be within reach to determine the best coefficient of the leading term $d^2$ in $a_d$. We pose this as an open problem.