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Keywords:

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

Abstract

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. MAIN RESULTS
  5. 3. PROOF OF THEOREM 1
  6. 4. PROOF OF THEOREM 2
  7. 5. PROOF OF THEOREM
  8. 6. PROOF OF THEOREM
  9. Appendix
  10. REFERENCES

The total domination number inline image 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, inline image. This bound is optimal in the sense that given any inline image, there exist graphs G with diameter 2 of all sufficiently large even orders n such that inline image.

1. INTRODUCTION

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. MAIN RESULTS
  5. 3. PROOF OF THEOREM 1
  6. 4. PROOF OF THEOREM 2
  7. 5. PROOF OF THEOREM
  8. 6. PROOF OF THEOREM
  9. Appendix
  10. REFERENCES

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 inline image be a graph with vertex set inline image of order inline image and edge set inline image of size inline image. The open neighborhood of a vertex inline image is inline image and the closed neighborhood of v is inline image. For a set inline image, its open neighborhood is the set inline image and its closed neighborhood is the set inline image. The degree of v is inline image. The minimum and maximum degree among the vertices of G is denoted by inline image and inline image, respectively. If the graph G is clear from the context, we simply write inline image, inline image, and inline image rather than inline image, inline image, and inline image, 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 inline image 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 inline image. 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 inline image is a TD-set in G if inline image. The total domination number of G, denoted by inline image, is the minimum cardinality of a TD-set of G. A TD-set of G of cardinality inline image is called a inline image-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 inline image is a finite set inline image of elements, called vertices, together with a finite multiset inline image 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 inline image of H is the minimum size of a transversal in H. A transversal of H of size inline image is called a inline image set.

For a graph inline image, we denote by inline image the open neighborhood hypergraph, abbreviated as ONH, of G; that is, inline image 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

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. MAIN RESULTS
  5. 3. PROOF OF THEOREM 1
  6. 4. PROOF OF THEOREM 2
  7. 5. PROOF OF THEOREM
  8. 6. PROOF OF THEOREM
  9. Appendix
  10. REFERENCES

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 inline image. For example, for inline image, the Cartesian product inline image is a diameter-2 graph of order inline image satisfying inline image. 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 inline image shown in Figure 1.

image

Figure 1. The graph G9 of order n with inline image.

Download figure to PowerPoint

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

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 inline image with inline image, then inline image.

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 inline image or inline image, then inline image.

As observed earlier, there exist diameter-2 graphs G of arbitrarily large order n such that inline image. In view of the above, it is a natural question to ask whether inline image 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 inline image, then there is a narrow range for the minimum degree δ of such a graph G, namely inline image.

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 inline image. 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 inline image.

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

3. PROOF OF THEOREM 1

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. MAIN RESULTS
  5. 3. PROOF OF THEOREM 1
  6. 4. PROOF OF THEOREM 2
  7. 5. PROOF OF THEOREM
  8. 6. PROOF OF THEOREM
  9. Appendix
  10. REFERENCES

Let inline image be a diameter-2 graph of order n. If v is an arbitrary vertex in G, then the diameter-2 constraint implies that inline image 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 inline image and exist for inline image 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 inline image vertices),
  2. the Petersen graph (3-regular graph on inline image vertices), and
  3. the Hoffman–Singleton graph (7-regular on inline image vertices).

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

4. PROOF OF THEOREM 2

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. MAIN RESULTS
  5. 3. PROOF OF THEOREM 1
  6. 4. PROOF OF THEOREM 2
  7. 5. PROOF OF THEOREM
  8. 6. PROOF OF THEOREM
  9. Appendix
  10. REFERENCES

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 inline image, then

  • display math

Recall the statement of Theorem 2.

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

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

  • math image

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

  • display math

This is clearly true when inline image as inline image is a decreasing function and when inline image it is below 1/2. It is easy to check that it is also true for all inline image (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'

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. MAIN RESULTS
  5. 3. PROOF OF THEOREM 1
  6. 4. PROOF OF THEOREM 2
  7. 5. PROOF OF THEOREM
  8. 6. PROOF OF THEOREM
  9. Appendix
  10. REFERENCES

Recall the statement of Theorem 4.

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

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

  • math image

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

  • display math

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

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. MAIN RESULTS
  5. 3. PROOF OF THEOREM 1
  6. 4. PROOF OF THEOREM 2
  7. 5. PROOF OF THEOREM
  8. 6. PROOF OF THEOREM
  9. Appendix
  10. REFERENCES

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 inline image.

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 inline image, there exists a constant inline image such that for all inline image there exists an intersecting uniform hypergraph H with inline image and inline image.

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

math image

We note that inline image. Now define

  • display math

and so inline image. Since inline image, we note that inline image. Recall that inline image, 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 inline image, we haveinline image.

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

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

  1. math image for all math image.
  2. math image for all inline image and all math image.
  3. math image for all inline image.
  4. math image for all inline image.

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

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

  • display math

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

Claim C. math image.

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

  • math image

that produces the desired result.

We now return to the proof of Theorem 8. Let

  • display math

We will now show that a random k-uniform hypergraph H on inline image vertices and with n edges has probability strictly less than inline image 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 inline image vertices and with n edges, all chosen at random (an edge may be chosen several times). Let inline image 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 inline image.

Claim D. inline image.

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

  • display math

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

  • display math

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

  • math image

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:

  • display math

Let

  • display math

and recall that inline image. Hence, the following holds:

  • math image

As observed earlier, inline image. Therefore, inline image, 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. inline image.

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

  • display math

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

  • math image

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

  • display math

Claim F. inline image.

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

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

  • math image

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 inline image, there exist diameter-2 graphs G of all sufficiently large even orders n satisfying inline image.

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

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

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

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

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

  • display math

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 inline image, there exist diameter-2 graphs G of all sufficiently large even orders n such that inline image.

Proof of Theorem 5. Given inline image, let c be any fixed number such that inline image, 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): inline image. Therefore the following holds:

  • math image

The desired result now follows from Theorem 9.

Appendix

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. MAIN RESULTS
  5. 3. PROOF OF THEOREM 1
  6. 4. PROOF OF THEOREM 2
  7. 5. PROOF OF THEOREM
  8. 6. PROOF OF THEOREM
  9. Appendix
  10. REFERENCES
Table A1. Comparison of selected values in the proof of Theorem 2
n math imageinline image
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 math imageinline image
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

REFERENCES

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. MAIN RESULTS
  5. 3. PROOF OF THEOREM 1
  6. 4. PROOF OF THEOREM 2
  7. 5. PROOF OF THEOREM
  8. 6. PROOF OF THEOREM
  9. Appendix
  10. REFERENCES
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