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### Keywords:

• total domination;
• transversals;
• hypergraphs;
• diameter-2

### Abstract

The total domination number of a graph G is the minimum cardinality of a set S of vertices, so that every vertex of G is adjacent to a vertex in S. In this article, we determine an optimal upper bound on the total domination number of a graph with diameter 2. We show that for every graph G on n vertices with diameter 2, . This bound is optimal in the sense that given any , there exist graphs G with diameter 2 of all sufficiently large even orders n such that .

### 1. INTRODUCTION

Our aim in this article is to obtain an optimal upper bound on the total domination number of a graph with diameter 2. Our proof techniques use an interplay between total domination in graphs and transversals in hypergraphs, as well as probabilistic methods.

For notation and graph theory terminology, we in general follow [4]. Specifically, let be a graph with vertex set of order and edge set of size . The open neighborhood of a vertex is and the closed neighborhood of v is . For a set , its open neighborhood is the set and its closed neighborhood is the set . The degree of v is . The minimum and maximum degree among the vertices of G is denoted by and , respectively. If the graph G is clear from the context, we simply write , , and rather than , , and , respectively. The girth of G is the length of a shortest cycle of G.

For two vertices u and v in a connected graph G, the distance between u and v is the length of a shortest uv path in G. The maximum distance among all pairs of vertices of G is the diameter of G, which is denoted by diam(G). We say that G is a diameter-2 graph if . Distance and diameter are fundamental concepts in graph theory and are well-studied in the literature.

A total dominating set, abbreviated as TD-set, of G is a set S of vertices of G such that every vertex is adjacent to a vertex in S. Thus, a set is a TD-set in G if . The total domination number of G, denoted by , is the minimum cardinality of a TD-set of G. A TD-set of G of cardinality is called a -set. Total domination in graphs is now well studied in graph theory. The literature on the subject has been surveyed and detailed in the recent book [9]. A survey of total domination in graphs can also be found in [6].

Hypergraphs are systems of sets that are conceived as natural extensions of graphs. A hypergraph is a finite set of elements, called vertices, together with a finite multiset of arbitrary subsets of V, called hyperedges or simply edges. A k-uniform hypergraph is a hypergraph in which every edge has size k. Every (simple) graph is a 2-uniform hypergraph. Thus, graphs are special hypergraphs. A hypergraph H is called an intersecting hypergraph if every two distinct edges of H have a nonempty intersection.

A subset T of vertices in a hypergraph H is a transversal in H if T has a nonempty intersection with every edge of H. A transversal is also called a hitting set in the literature. The transversal number of H is the minimum size of a transversal in H. A transversal of H of size is called a set.

For a graph , we denote by the open neighborhood hypergraph, abbreviated as ONH, of G; that is, is the hypergraph with vertex set V and with edge set C consisting of the open neighborhoods of vertices of V in G.

### 2. MAIN RESULTS

The decision problem to determine the total domination number of a graph remains NP-hard (non-deterministic polynomial-time hard) even when restricted to cubic graphs or planar graphs of maximum degree 3 [2]. Hence, it is of interest to determine upper bounds on the total domination number of a graph. Bounds on the total domination number of a graph in terms of its order and minimum degree can be found, for example, in [1, 5, 7, 8, 12]. Goddard and Henning [3] showed that the total domination number of a diameter-2 planar graph is at most 3. However, there exist diameter-2 nonplanar graphs G of arbitrarily large order n such that . For example, for , the Cartesian product is a diameter-2 graph of order satisfying . In this article, we determine an optimal upper bound on the total domination number of a diameter-2 graph.

We first establish some elementary properties of diameter-2 graphs. We note that if G is a diameter-2 graph, then G is either a star or G has girth 3, 4, or 5. Let G9 be the diameter-2 graph of order shown in Figure 1.

A proof of Theorem 1 is given in Section 'PROOF OF THEOREM 1'.

Theorem 1. Let G be a diameter-2 graph of order n. Then the following hold.

1. .
2. If , then with equality if and only if .
3. If G has girth 5, then .

We next establish an upper bound on the total domination number of a general graph with large minimum degree. A proof of Theorem 2 is given in Section 'PROOF OF THEOREM 2'.

Theorem 2. If G is a graph of order with , then .

Theorems 1(a) and 2 imply the following result.

Corollary 3. If G is a diameter-2 graph of order n with minimum degree δ, where or , then .

As observed earlier, there exist diameter-2 graphs G of arbitrarily large order n such that . In view of the above, it is a natural question to ask whether is an upper bound on the total domination number of a diameter-2 graph of order n. By Corollary 3, if there do exist diameter-2 graphs G of order n for which , then there is a narrow range for the minimum degree δ of such a graph G, namely .

We have two main aims in this article. First to provide an optimal upper bound on the total domination number of a diameter-2 graph of order n. Second to show that there do exist diameter-2 graphs of arbitrarily large order n such that . More precisely, we will prove the following two results. Proofs of Theorem 4 and Theorem 5 can be found in Section 'PROOF OF THEOREM 'PROOF OF THEOREM 2'' and Section 'PROOF OF THEOREM 'PROOF OF THEOREM 'PROOF OF THEOREM 2''', respectively.

Theorem 4. If G is a diameter-2 graph of order n, then .

Theorem 5. Given any , there exist diameter-2 graphs G of all sufficiently large even orders n such that .

### 3. PROOF OF THEOREM 1

Let be a diameter-2 graph of order n. If v is an arbitrary vertex in G, then the diameter-2 constraint implies that is a TD-set in G. In particular, choosing v to be a vertex of minimum degree proves part (a) of Theorem 1.

Theorem 1(b) is a result for small n that can readily be checked by computer (or see [13] for a mathematical proof).

To prove part (c) of Theorem 1, we remark that the diameter-2 graphs of girth 5 are precisely the diameter-2 Moore graphs. It is shown (see [10, 11]) that Moore graphs are r-regular and that diameter-2 Moore graphs have an order and exist for and possibly 57, but for no other degrees. The Moore graphs for the first three values of r are unique, namely,

1. the 5-cycle (2-regular graph on vertices),
2. the Petersen graph (3-regular graph on vertices), and
3. the Hoffman–Singleton graph (7-regular on vertices).

Since G is a diameter-2 graph of girth 5, the graph G is a diameter-2 Moore graph and . Hence, . Let D be a -set. Then, , implying that ; or equivalently, . Therefore by Part (a), we have that , or, equivalently, . Since both and are integers, . This establishes part (c) and completes the proof of Theorem 1.

### 4. PROOF OF THEOREM 2

Before we present a proof of Theorem 2, we will need the following upper bound on the total domination number of a graph in terms of its minimum degree and its order that is established in [8].

Theorem 6. (([8])) If G is a graph of order n with minimum degree , then

Recall the statement of Theorem 2.

Theorem 2. If G is a graph of order with , then .

Proof of Theorem 2. Let G be a graph of order with minimum degree . If , then . Hence, we may assume that . Therefore, since , we note that . Applying Theorem 6 to the graph G and noting that is a decreasing function in δ, we have that

To prove that , it therefore suffices for us to show that

This is clearly true when as is a decreasing function and when it is below 1/2. It is easy to check that it is also true for all (by using a computer). A few selected values can be found in Table A1 in Appendix. This completes the proof of Theorem 2.

### 5. PROOF OF THEOREM 'PROOF OF THEOREM 2'

Recall the statement of Theorem 4.

Theorem 4. If G is a diameter-2 graph of order , then .

Proof of Theorem 4. Let G be a diameter-2 graph of order with minimum degree δ. If , then by Theorem 1(a). Hence we may assume that . In particular, we note that . Applying Theorem 6 to the graph G and noting that is a decreasing function, we have that

To prove that , it therefore suffices for us to show that

This is clearly true when as is a decreasing function and when , it is below 1/2. It is easy to check that it is also true for all , and the values are given in Table A2 in Appendix. This completes the proof of Theorem 4.

### 6. PROOF OF THEOREM 'PROOF OF THEOREM 'PROOF OF THEOREM 2''

In order to prove Theorem 5, we use an interplay between total domination in graphs and transversals in hypergraphs. Perhaps much of the recent interest in total domination in graphs arises from the fact that total domination in graphs can be translated to the problem of finding transversals in hypergraphs. The following observation, first observed by Thomassé and Yeo [12], shows that the transversal number of the ONH of a graph is precisely the total domination number of the graph.

Observation 7. (([12])) If G is a graph with no isolated vertex, then .

We first prove the following result on the existence of intersecting uniform hypergraphs with large transversal number. Our proof is motivated in part by Theorem 6 in [12].

Theorem 8. For any constant , there exists a constant such that for all there exists an intersecting uniform hypergraph H with and .

Proof. Let be a given constant, and define the two constants and by

We note that . Now define

and so . Since , we note that . Recall that , where e is the base of the natural logarithm. Hence we have the following well-known result.

Claim A. The following holds.

1. For all , we have.

2. Given , where , there exists a constant such that , when .

We now define to be a constant that is sufficiently large for Conditions (a), (b), (c), and (d) below to hold.

1. for all .
2. for all and all .
3. for all .
4. for all .

We remark that Condition (a) is possible by Claim A(b) since defining such that , we have by Claim A(b) that for a sufficiently large, . We also remark that Condition (b) can clearly be satisfied as grows faster than any poly-log function in k and . Further Condition (c) can clearly be satisfied as is a positive constant, while Condition (d) can clearly be satisfied as is an increasing function that tends to infinity.

From the remarks above, the constant is well-defined. Let be arbitrary. By Condition (c) above, we have that , and so there exists an integer r such that . Let be defined such that . In particular, we note that and is an integer. Let

and note that . By our choice of , we have that k is an integer. We also remark that since , Condition (d) implies that . We proceed further with the following claim.

Claim C. .

Proof of Claim C. Since is an increasing function and since , we note that . Hence, Condition (a) implies that the following holds:

that produces the desired result.

We will now show that a random k-uniform hypergraph H on vertices and with n edges has probability strictly less than of having a transversal of size less than z. Furthermore, we will show that it has probability at least 1/2 of being intersecting, which implies that there exists an intersecting k-uniform hypergraph on n vertices and with n edges, which does not contain a transversal of size at most z. We will finally show that this implies that the theorem holds.

Let H be a hypergraph, with vertices and with n edges, all chosen at random (an edge may be chosen several times). Let denote the resulting edges. The following claim shows that the probability that H contains a transversal of size at most z is strictly less than .

Claim D. .

Proof of Claim D. Let Z be vertices chosen at random (a vertex may be picked several times). There are possible ways of choosing Z. For each , we have that

as each vertex in Z has probability of not lying in . Hence, the probability that for all , is the following (as the edges are chosen independently):

Let contain all possible Z’s (i.e., , and if and only if Z contains exactly elements, some of which may be identical). As was chosen with equal probability, namely , and since , we get the following:

As the right-hand side of above is an increasing function in z, we note that the probability that H has a transversal of size at most z is given by the following:

Let

and recall that . Hence, the following holds:

As observed earlier, . Therefore, , and the proof of Claim D is complete.

We show next that the probability that H is not intersecting is less than 1/2.

Claim E. .

Proof of Claim E. For , the probability that the edges and do not intersect (i.e., they are vertex disjoint) is the following:

Hence, by Claim A(a) and since , the above implies the following:

Since the edges and above were chosen arbitrarily and there are distinct pairs of edges in H, this implies the following bound on the probability that H is not an intersecting hypergraph. (Recall that .)

Claim F. .

Proof of Claim F. Let A be the event that and let B be the event that H is not intersecting. By Claim D, we have , while by Claim E we have , and so . Let and denote the complement of events A and B, respectively. Then, .

By Claim F, there exists an intersecting k-uniform hypergraph on vertices (and with n edges) such that . Therefore, the following holds:

This completes the proof of Theorem 8.

We next use an interplay between total domination in graphs and transversals in hypergraphs to establish the following graph-theoretic result.

Theorem 9. For any constant , there exist diameter-2 graphs G of all sufficiently large even orders n satisfying .

Proof. Let be an arbitrary given constant. It suffices us to show that there exists a constant such that for all even there exists a diameter-2 graph G of order n with .

Let , and so . By Theorem 8, there exists a constant such that for all there exists an intersecting k-uniform hypergraph H on vertices with . Let and let be an arbitrary even integer. Let , and so . By Theorem 8, let H be an intersecting k-uniform hypergraph on vertices and edges with .

Consider the incidence bipartite graph of the hypergraph H with partite sets and and where there is an edge between and if and only if x belongs to the hyperedge y in H. Let G be obtained from by adding all edges joining vertices in X. We note that Y is an independent set in G. Further, G has order .

We first show that G has diameter 2. If , then since is a clique. Suppose and . If x belongs to the edge y in H, then . Otherwise, let be any vertex in the edge y in H. Since is a clique, , implying that . Finally suppose that , and so x and y are both hyperedges in H. Since H is intersecting, there is a vertex u in H that belongs to both hyperedges x and y. Thus in the graph G, we have that and that u is adjacent to both x and y in G. This implies that is a path in G, and so . Therefore, G is a diameter-2 graph.

It remains for us to provide a lower bound on . Let S be a -set. Since Y is an independent set, the set totally dominates the set Y in G. By construction of the graph G this implies that is a transversal in H. Therefore, . Hence, since and , we have

This completes the proof of Theorem 9.

We are now in a position to prove Theorem 5. Recall the statement of the theorem.

Theorem 5.    Given any , there exist diameter-2 graphs G of all sufficiently large even orders n such that .

Proof of Theorem 5. Given , let c be any fixed number such that , and note that the following holds when n is large enough (as the left-hand side tends to infinity when n gets large and the right-hand side is a constant): . Therefore the following holds:

The desired result now follows from Theorem 9.

### Appendix

Table A1. Comparison of selected values in the proof of Theorem 2
n
31.724851.86603
41.913932
52.050522.11803
62.164372.22474
72.26482.32288
82.355982.41421
92.440162.5
102.518792.58114
112.592822.65831
122.662982.73205
203.130763.23607
303.581693.73861
504.273084.53553
1005.48776
2007.119738.07107
Table A2. Comparison of values in the proof of Theorem 4
n
21.387361.58871
31.730191.90772
41.976062.17741
52.181972.41838
62.363542.6394
72.527982.84536
82.67943.03933
92.820473.22346
102.953023.39926
113.078423.56792
123.197673.73033
133.311593.88723
143.42084.03919
153.525844.18672
163.627134.33022
173.725034.47004
183.819864.60648
193.911874.7398
204.00134.87023
214.088344.99797
224.173175.12319
234.255945.24607

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