The domination number of round digraphs

Abstract The concept of the domination number plays an important role in both theory and applications of digraphs. Let D = ( V , A ) D=(V,A) be a digraph. A vertex subset T ⊆ V ( D ) T\subseteq V(D) is called a dominating set of D, if there is a vertex t ∈ T t\in T such that t v ∈ A ( D ) tv\in A(D) for every vertex v ∈ V ( D ) \ T v\in V(D)\backslash T . The dominating number of D is the cardinality of a smallest dominating set of D, denoted by γ ( D ) \gamma (D) . In this paper, the domination number of round digraphs is characterized completely.


Introduction
The domination theory of graphs was derived from a board game in ancient India. In 1962, Ore formally gave the definitions of the dominating set and the domination number in [1]. Due to the universality of its applications to both theoretical and practical problems, domination has become one of the important research topics in graph theory. A summary of most important results and applications can be found in [2]. Problems of resource allocations and scheduling in networks are frequently formulated as domination problems on underlying graphs (digraphs). By contrast, domination in digraphs has not yet gained the same amount of attention, although it has several useful applications as well. For example, it has been used in the study of answering skyline query in the database [3] and routing problems in networks [4]. The relationship among domination numbers of different orientations of a graph was studied in [5]. The relevant results about domination numbers of digraphs can be found in [6][7][8][9][10]. Recent studies on domination theory include [11][12][13].
We refer the reader to [14] for terminology and notation not defined in this paper. Let = ( ) D V A , be a digraph, which means that V and A represent the vertex set and the arc set of D, respectively. The order of D is the number of vertices in D, denoted by | ( )| V D . If uv is an arc, then we say that u dominates v (or v is dominated by u) and use the notation → u v to denote this. For a vertex v of a digraph D, we define the vertex set ( ) = { ∈ { }| ∈ } We also call the vertex set ( ) ( ) be a vertex labelling of D. If there is always < i j for every arc v v i j in D, then we often refer to the vertex labelling as an acyclic ordering of D. A walk in D is an alternating sequence of vertices v i and arcs a j from D such that the tail of a i is v i and the head of a i is , and open otherwise. The set of vertices , -walk and a ( ) v u , -walk. A semicomplete digraph is a digraph in which every pair of distinct vertices is adjacent. A tournament is a semicomplete digraph with no cycle of length two. A digraph D is locally in-semicomplete (out-semicomplete) if, for every vertex x of D, the in-neighbours (out-neighbours) of x induce a semicomplete digraph. A digraph D is locally semicomplete if it is both locally in-semicomplete and locally out-semicomplete. A locally semicomplete digraph with no 2-cycle is a local tournament. If a digraph is a locally semicomplete digraph (local tournament) but not a semicomplete digraph (tournament), then the digraph is a purely local semicomplete digraph (purely local tournament). Related surveys about the locally semicomplete digraphs can be found in [15,16].
A digraph on n vertices is round if we can label its vertices , be a digraph, and let T be a subset of the vertices of D. If for every vertex ∈ ( ) v V D T \ , there is a vertex ∈ t T such that ∈ ( ) tv A D , then we say that T is a dominating set of D and denote it by → T D. The dominating number of D is the cardinality of a smallest dominating set of D, denoted by ( ) γ D . We need the following lemma and theorem in order to prove the main theorems.  In this paper, the domination number of a round digraph is characterized by studying the round local tournament and the round non-local tournament, respectively.
0,1, , 1 , then the arc v v i j is called a cross arc on P n . If there is no cross arc v v i j α α on P n such that < < < i i j j α α , then the cross arc v v i j is called a maximal cross arc on P n . We call the vertex set covered by the maximal cross arc v v i j . We call the set G a maximal cross-arc chain on P n , if there is a maximal cross-arc set t is a maximal cross arc on P n , ∈ { … − }} t k 0, 1, , 1 on P n satisfying one of the following conditions: β β are two cross arcs on P n ; or (b) v v i j 0 0 is a maximal cross arc on P n . There is a maximal cross arc , and there is no cross arc Figure 1(a)).
, and when ≠ ( ; and (c) There is no maximal cross arc v v i j γ γ on P n such that 2 . In addition to the invalid cross arcs in G, the remaining maximal cross arcs are called the valid cross arcs of G. Let P m be a subpath of P n . If all the vertices on P m are not covered by any maximal cross-arc chain, then P m is called a pure subpath of P n . If there is no is a pure subpath of P n , then call P m a maximal pure subpath of P n and all vertices in ( ) V P m covered by P m . Figure 1 illustrates these definitions.
Subsequently, we show the partition problem of the vertices of a round purely local tournament which is non-strong. Let D be a round purely local tournament which is non-strong. Let 1 is the only Hamilton path in D. If there exists a cross arc on P, then it can only be forward arc (that is, from the vertex with a small subscript to the vertex with a large subscript). Thus, ( ) V D can form a partition. It means where B i is covered by either some maximal pure subpath or some maximal cross-arc chain ( Figure 2).
According to the partition about the vertex set above, the following conclusions can be obtained.    are all maximal pure subpaths on P; is an only maximal cross-arc chain on path = 14 15 16 are all maximal pure subpaths on P.
Proof. Since P m is a directed path, . Therefore, M is not a dominating set of P m . For the arbitrariness of M, we have Let D be a round purely local tournament which is non-strong on n vertices. Let where τ is the number of all valid arcs in G, and all subscripts are ordered in the round ordering.
Proof. Now we distinguish two cases to prove this lemma.

Case 1. There is no invalid cross arc in G.
According to the definitions of the maximal cross arc and the round digraph, there must be . Thus, there is at least  (Figure 3(a)).

Case 2.
There is at least one invalid cross arc in G.
Let v v i j α α be any invalid cross arc of G. According to the definition of an invalid cross arc, is covered by the maximal cross arc + + v v i j α α 1 1 . By the definitions of the cover and the round digraph, we . For the arbitrariness of v v i j α α , all vertices covered by the invalid cross arcs in G can be covered by the valid cross arcs in G. This means ( 〈 〉) ≤ γ D B τ, where τ is the number of valid arcs in G. According to Case 1, τ can be proved similarly. Then, ( 〈 〉) = γ D B τ (Figure 3(b)).
be a maximal subpath on P, and . According to the proof of Lemma 2.2 and the structure of the round purely local tournament which is non-strong, the minimum dominating set of 〈 〉 , , , , , By the proof of Case 1 in Lemma 2.2, there must be Case 2. There is a maximal pure subpath P β such that According to Lemma 2.1, there must be a vertex set { } ⊆ ( ) Therefore, M is not a dominating set of D anyway. By the arbitrariness of M, Thus, t is a maximal cross arc on the cycle C n , ∈ { … − }} t k 0, 1, , 1 on C n satisfied: (1) For = k 1, (a) v v i j 0 0 is a maximal cross arc onC n and there is no set{ ( Figure 5(b)).
If there is a maximal cross arc , then the maximal cross arc v v i j τ τ is an invalid cross arc of G, where ≤ ≤ − τ k 1 2 . In addition to the invalid cross arcs in G, we call the remaining maximal cross arcs the valid cross arcs of G. Let P m be a subpath on C n . If all the vertices on P m are not covered by any maximal cross-arc chain, then call P m a pure subpath on C n . If there is no vertex is a pure subpath on C n , then P m is a maximal pure subpath of C n and ( ) V P m is covered by P m . Figure 5 illustrates these definitions.
Next, we show the partition problem of the vertices of a round purely local tournament which is strong. Let D be a round purely local tournament which is strong. Let

The domination number of a round non-local tournament
If a round digraph D is a non-local tournament, then there is a 2-cycle in D by Lemma 1.1. It implies D is a semicomplete digraph or a purely local semicomplete digraph. According to the definition of the round digraph, one can know that D is strongly connected.