On Total Domination in Graphs


 Antonia Ashlynn Stephens
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1 University of Houston  Downtown Senior Project  Fall 2012 On Total Domination in Graphs Author: David Amos Advisor: Dr. Ermelinda DeLaViña Senior Project Committee: Dr. Sergiy Koshkin Dr. Ryan Pepper Department Chair: Dr. Shishen Xie December 7, 2012
2 Contents 1 Graph Theory Background The definition of a graph Neighborhoods and degree Distance, radius, and diameter Domination and total domination Independence and matchings Some special families of graphs Characterizations of total domination number A Short Survey of Known Results Some basic results for domination and total domination Total domination and minimum degree Total domination in trees Total domination in planar graphs Conjectures of Graffiti.pc Total domination and maximum degree Total domination and cut vertices Total domination and degree two vertices Total Domination and G Miscellaneous conjectures Concluding Remarks 33
3 List of Figures 2.1 The graphs G 10 (left) and H 10 from Theorem The unique planar graph with diameter 2 and γ = Sharpness of Corollary and Theorem A smallest counterexample to Conjecture with order A 19vertex smallest counterexample to Conjecture Sharpness of Proposition Illustration of the argument in Proposition Sharpness of Proposition vertex counterexample to γ t (G) N(V 2 ) V Sharpness of Theorem The graph Ŝ A 16vertex counterexample to Conjecture A 21vertex counterexample to Conjecture An 18vertex counterexample to Conjecture A 24vertex counterexample to Conjecture A 14vertex counterexample to Conjectures and A 13vertex counterexample to Conjecture
4 Abstract Let G = (V, E) be a finite, simple, undirected graph. A set S V is called a total dominating set if every vertex of V is adjacent to some vertex of S. Interest in total domination began when the concept was introduced by Cockayne, Dawes, and Hedetniemi [6] in In 1998, two books on the subject appeared ([11] and [12]), followed by a survey of more recent results in 2009 [15]. The total domination number of a graph G, denoted γ t (G) is the cardinality of a smallest total dominating set. The problem of computing γ t (G) for a given graph was shown to be NPcomplete by Pfaff et al. [20]. However, a linear algorithm exists for trees [18]. The overall complexity of calculating γ t (G) is the primary motivation for discovering bounds, preferably in terms of easily computable invariants. For this project, DeLaViña s conjecturing program Graffiti.pc was used to investigate lower and upper bounds for the total domination number of a graph. We survey a few known results on total domination and discuss the conjectures of Graffiti.pc that were resolved during the project. Proofs are supplied for true conjectures and counterexamples are given for false conjectures. In some cases, smallest counterexamples are provided.
5 Acknowledgements I would like to thank Dr. Ermelinda DeLaViña for her interest in me as a student, and for taking me under her wing. Her advice was always wise, her support, unfaltering, and her careful examination of my logic much appreciated. I am also indebted to Dr. Ryan Pepper, whose enthusiasm is infectious. His Texas Style graph theory class inspired me to pursue a senior project in graph theory. I am also extremely grateful for the mentorship of Dr. Sergiy Koshkin, who introduced me to graph theory during his discrete mathematics class. The directed study in advanced linear algebra I took with Dr. Koshkin in the summer of 2011 was one of the most impactful experiences during my time at UHD. I consider that directed study to be my first real introduction to higher mathematics, and it could not have come from a better person. Finally, I would like to thank Dr. Edwin Tecarro for teaching the senior project class (and also for introducing me to real analysis), and the other faculty members in the Computer and Mathematical Sciences Department at the University of Houston  Downtown. The have all helped to make my experience at UHD absolutely wonderful.
6 Chapter 1 Graph Theory Background For the most part, standard notation is adopted. Special definitions and notation are introduced at the appropriate time. Following are detailed definitions of the basic concepts used throughout this report. A few basic results are also included. 1.1 The definition of a graph Let G = (V, E) be a graph with vertex set V and edge set E. If V is finite, we say that G is a finite graph. The order of a graph G, denoted n, is simply V, and we denote the size of a graph by m = E. Two vertices u, v V are said to be adjacent, denoted u v, if there exists an edge e E whose endpoints are u and v, in which case we denote the edge by uv (or by vu, the order does not matter). If v V is the endpoint of some edge e E, then e is said to be incident to v. For any u V, the edge uu is called a loop. For some u, v V, the edge uv is called a multiedge if it appears in E more than once. A graph whose edge set contains no loops and no multiedges is called simple. All graphs considered in this report are simple and finite. A graph of order one is said to be trivial, and we will often exclude such a graph from our discussion. Any graph whose edge set is the empty set is called an empty graph. A path of length k is a set of vertices {x 1, x 2,..., x k } such that x 1 x 2, x 2 x 3,..., x k 1 x k. x 1 and x 2 are called the endpoints of the path. We will often refer to a path by the endpoints. For example, a path whose endpoints are u and v is called a uv path. A cycle is a path with the added condition that x k x 1. If there exists a path between every pair of vertices of a graph, then the graph is said to be connected. Otherwise, the graph is disconnected. The connected parts of a disconnected graph are called the components of the graph. For a set X V, the subgraph induced by X, denoted G[X], is the graph with vertex set X and edge set E = {uv E(G) : u, v X}. 1
7 1.2 Neighborhoods and degree The open neighborhood of a vertex v V is the set N(v) = {u V : u v}. The closed neighborhood of v is the set N[v] = N(v) {v}. If N(v) =, then v is said to be isolated. An isolated vertex is sometimes called an isolate. A vertex u N(v) is called a neighbor of v. It is often useful to consider the neighborhoods of an entire set of vertices. For some X V (G) we call the set N(X) = v X N(v) the open neighborhood of X. The closed neighborhood of X is the set N[X] = N(X) X. The degree of a vertex v V is N(v) and is denoted d G (v) (or simply d(v) when the context is clear). We let δ(g) = min{d(v) : v V } and (G) = max{d(v) : v V } denote the minimum and maximum degrees respectively (if the context is clear, we will simply let δ = δ(g) and = (G)). A vertex of degree one is called a pendant, and its neighbor is called a support vertex. Many of the conjectures produced by Graffiti.pc involve special subsets of V. For compactness, we let V k = {v V : d(v) = k}. Hence V δ is the set of minimum degree vertices, and V 2 is the set of degree two vertices. 1.3 Distance, radius, and diameter The distance between two vertices u and v, denoted d(u, v), is the length of a shortest path whose endpoints are u and v. If no uv path exists, it is customary to let d(u, v) =. The eccentricity of a vertex v is the maximum distance from v to any other vertex of the graph. The maximum and minimum eccentricities, over all vertices, are referred to respectively as the diameter and radius of the graph. The diameter of a graph is denoted diam(g) and the radius is denoted rad(g). 1.4 Domination and total domination A set D V is called a dominating set if every vertex in V D is adjacent to some vertex of D. Notice that D is a dominating set if and only if N[D] = V. The domination number of G, denoted γ = γ(g), is the cardinality of a smallest dominating set of V. We call a smallest dominating set a γset. A set S V is a total dominating set if every vertex in V is adjacent to some vertex of S. Alternatively, we may define a dominating set D to be a total dominating set if G[D] has no isolated vertices. The total domination number of G, denoted γ t = γ t (G), is the cardinality of a smallest total dominating set, and we refer to such a set as a γ t set. An immediate consequence of the definitions of domination number and total domination number is that, for any graph G, γ t γ. 2
8 1.5 Independence and matchings For any set I V, if G[I] is empty, then I is called an independent set. The independence number of a graph G, denoted α = α(g) is the cardinality of a largest independent set. There is a basic relation between independence number and domination number. Proposition Let G be a graph. Then, α γ. Proof. Let I V (G) be a largest independent set. Then for any v V I, the subgraph induced by I {v} has at least one edge uv for some u I, since otherwise, if such a v existed, the set I {v} would be a larger independent set. Thus, every v V is either in I or adjacent to a vertex of I, from which it follows that I is also a dominating set. Since I = α, we have α γ. The concept of independence may be extended to edges by defining two edges to be independent if they have no common endpoint. A matching is a set M E with the property that the edges in M are pairwise independent. The matching number of G is the cardinality of a largest matching and is denoted µ = µ(g). Endpoints of a matching edge are called saturated vertices. An unsaturated vertex is any vertex that is not an endpoint of a matching edge. It is easy to see that µ can be at most half the order of G. Proposition Let G be a graph of order n. Then, µ n 2. Proof. The largest possible matching for any graph could at most saturate every vertex. Since every edge has two vertices, it follows that 2µ n. The proposition follows immediately. 1.6 Some special families of graphs The path on n vertices is denoted P n, and is the graph with V (P n ) = {v 1, v 2,..., v n } and E(P n ) = {v 1 v 2, v 2 v 3,..., v n 1 v n }. The cycle on n vertices is denoted C n and has V (C n ) = V (P n ) and E(C n ) = E(P n ) {v n v 1 }. A graph is complete if every vertex is adjacent to every other vertex of the graph. We denote the complete graph with order n by K n. For any graph G, a set of vertices is called a clique if the subgraph induced by the set is complete. The clique number of a graph is the cardinality of a largest clique and is denoted ω = ω(g). If the vertex set of a graph can be partitioned into two disjoint independent sets A and B, then the graph is called bipartite. Every edge in a bipartite graph has the form ab where a A and b B. If every possible edge between the sets A and B is present, then the graph is called complete bipartite and denoted K n1,n 2 where n 1 = A and n 2 = B. The star on n vertices, denoted S n is, is equivalent to K 1,n 1. The degree n 1 vertex of S n is called the center. A binary star is made up of two stars, with an edge added between the centers. 3
9 The kcorona of a connected graph H, denoted H P k, is a the graph formed as follows: Take n(h) copies of P k and let X be the set made up of exactly one endpoint from each copy of P k. Let f : V (H) X be a bijection. Connect each v V (H) to an endpoint of a corresponding P k with the edge vf(v). 1.7 Characterizations of total domination number Total domination number is easily computed for several families of graphs. First, we make the following observation. Observation Let G be a graph of order n with no isolated vertices. If n 2, then γ t = 2. Graphs for which Observation applies include stars and complete graphs. Other graphs with γ t = 2 include binary stars and complete bipartite graphs. In general, however, the problem of calculating the total domination number in bipartite graphs remains difficult [20]. Total domination number is easily calculated for cycles and paths. 4
10 Proposition The total domination number of a cycle C n or a path P n on n 3 vertices is given by: (a) γ t (C n ) = γ t (P n ) = n 2 if n 0(mod 4). (b) γ t (C n ) = γ t (P n ) = n+2 2 if n 2(mod 4). (c) γ t (C n ) = γ t (P n ) = n+1 2 otherwise. Characterizations of total domination number are known for several other families of graphs (see, for example, Henning s survey [15]). However, the above characterizations will be the most useful to us for the remainder of this report. 5
11 Chapter 2 A Short Survey of Known Results In this chapter, we survey some of the known results for total domination. First, we examine the relationship between domination and total domination. 2.1 Some basic results for domination and total domination The following basic upper bound for domination number is considered to be folklore among graph theorists. The proof is provided for completeness. Theorem Let G be a graph with no isolated vertices. Then γ n 2. Proof. Let D V (G) be a γset. Since G has no isolated vertices, every v D has at least one neighbor in V D. This means that V D is also a dominating set. If D > n/2, then V D is a smaller dominating set, contradicting the choice of D as a γset. Thus γ = D n/2. Bollobás and Cockayne proved a useful property of γsets in [1]. Theorem (Bollobás and Cockayne [1]). Every graph G with no isolated vertices has a γset D such that every v D has the property that there exists a vertex v V D that is adjacent to v but to no other vertex of D. The vertex v from Theorem is called a private neighbor of v. From now on, we will assume that every γset is chosen to be one whose existence is guaranteed by Theorem This allows us to prove the following triple inequality. Theorem Let G be a graph with no isolated vertices. Then γ γ t 2γ. 6
12 Proof. The first inequality follows immediately from the definitions of domination number and total domination number. Let D be a γset. If D is not a total dominating set, it is because the subgraph induced by D has isolated vertices. By Theorem 2.1.2, each of these isolated vertices has a private neighbor. To construct a total dominating set, we simply add to D a private neighbor for each isolated vertex, and call the new set D. At most D private neighbors could have been added to form D. That is, D 2 D. Since D is a total dominating set, D γ t. Therefore, γ t 2 D = 2γ. This proves the second inequality. Finally, we mention a basic lower bound for total domination number. Theorem Let G be a graph of order n with no isolated vertices. Then γ t n. Proof. Let S be a γ t set of G. Then, by definition, every vertex of G is adjacent to some vertex of S. That is, N(S) = V (G). Since every v S can have at most neighbors, it follows that γ t V = n. The theorem follows by dividing this inequality by. In section 3.1, we prove a few corollaries to Theorem Total domination and minimum degree Cockayne et al. [6] proved the following for connected graphs. Theorem (Cockayne et al. [6]). Let G be a connected graph of order n 3 with no isolated vertices. Then γ t 2 3 n. Equality for Theorem was characterized by Brigham, Carrington, and Vitray [2]. Theorem (Brigham, Carrington, and Vitray [2]). Let G be a connected graph of order n 3. Then γ t = 2n/3 if and only if G is C 3, C 6 or H P 2 for some connected graph H. Sun [21] showed that the bound in Theorem can be improved upon for connected graphs with minimum degree at least two. Theorem (Sun [21]). Let G be a connected graph of order n and δ 2. Then γ t 4 7 (n + 1). Henning [14] improved this bound slightly by disregarding six graphs with small orders. The graphs G 10 and H 10 are displayed in Fig Theorem (Henning [14]). If G {C 3, C 5, C 6, C 10, G 10, H 10 } is a connected graph of order n and δ 2, then γ t 4 7 n. Chvátal and McDiarmid [4] and Tuza [24] independently proved a theorem concerning transversals in hypergraphs. Theorem is a consequence of their result. 7
13 Figure 2.1: The graphs G 10 (left) and H 10 from Theorem Theorem (Chvátal and McDiarmid [4]; Tuza [24]). Let G be a graph of order n and δ 3. Then γ t n 2. The following result for graphs with minimum degree at least four is a consequence of a result of Thomassé and Yeo [22] for hypergraphs. Theorem (Thomassé and Yeo [22]). Let G be a graph of order n and δ 4. Then γ t 3 7 n. 2.3 Total domination in trees A graph T is called a tree if it has no cycles. In trees, a vertex of degree one is called a leaf. Trees are of interest for several reasons, but in particular, every connected graph G has a spanning tree whose vertex set is V (T ) and whose edge set is a subset of E(T ). Furthermore, every connected graph has a spanning tree whose total domination number is the same as the graph (see Lemma 2 in [8]). In some cases, it is possible to say something about total domination in a graph by considering only one of its spanning trees. This motivates finding sharp bounds for the total domination number of trees, even though, as mentioned in the abstract, a linear algorithm exists for computing the total domination number of trees [18]. Another interesting aspect of trees, with respect to total domination, is that it is possible to characterize which vertices are in every γ t set or are in no γ t  set [5]. For example, support vertices must be in every γ t set. Furthermore, Haynes and Henning found three conditions that are equivalent to a tree having a unique minimum γ t set [13]. We mention a couple of results that will be of use later. Chellali and Haynes [3] related the total domination number of a tree to the order and the number of leaves of the tree. Theorem (Chellali and Haynes [3]). Let T be a nontrivial tree of order n and l leaves. Then γ t n+2 l 2. 8
14 We show in section 3.2 that Theorem is a corollary of one of Graffiti.pc s conjectures. With respect to the degree two vertices of a tree, DeLaViña et al. proved the following in [7]. Theorem (DeLaViña et al. [7]). Let T be a nontrivial tree. Let c 2 be the number of components of the subgraph induced by V (T ) V 2, and let p 2 be the order of a largest path in the subgraph induced by V 2. Then γ t c 2 + p Theorem is also a corollary of one of Graffiti.pc s conjectures, as will be seen in section Total domination in planar graphs A planar graph is a graph that can be drawn in the plane with no edge crossings. In his 2009 survey, Henning lists twenty fundamental problems in the study of total domination [15]. Problem 12 simply says Investigate total domination in planar graphs. Originally, the intent of this project was to do just this. However, planarity was never needed as a sufficient condition for proving the proposed conjectures of Graffiti.pc. Despite this, we mention a few known results for planar graphs. It is relatively simple to see that a tree with radius two and diameter four can have arbitrarily large domination number (consider the 1corona of stars, for instance). However, MacGillivray and Seyffarth [19] have shown that domination number is bounded in planar graphs with diameter at most three. Theorem (MacGillivray and Seyffarth [19]). If G is a planar graph with diameter two, then γ 3. Theorem (MacGillivray and Seyffarth [19]). Every planar graph with diameter three has domination number at most 10. Interestingly, there is a unique planar graph with diameter two and domination number three. This graph, displayed in Figure 2.2, was found by MacGillivray and Seyffarth [19]. Since γ t 2γ, by Theorem 2.1.3, then total domination number is also bounded in graphs with diameter two and three. Dorfling, Goddard, and Henning [9] proved the following: Theorem (Dorfling, Goddard, and Henning [9]). Let G be a planar graph with no isolated vertices. Then the following hold: (a) If diam(g) = 2, then γ t 3. (b) If diam(g) = 3 and rad(g) = 2, then γ t 5. (c) If diam(g) = 3 and rad(g) = 3, then γ t 10. (d) If diam(g) = 3 and G has sufficiently large order, then γ t 7. 9
15 Figure 2.2: The unique planar graph with diameter 2 and γ = 3. As pointed out by Dorfling, Goddard, and Henning in [9], it follows from Theorem 2.4.3(c) and (d) that there are finitely many planar graphs with diameter three and total domination number more than seven. At this time, it is unknown if any exist. Total domination in planar graphs continues to be a subject of new research. 10
16 Chapter 3 Conjectures of Graffiti.pc Unless explicitly stated otherwise, from this point on we will only consider graphs with no isolated vertices. We also assume that every γset is chosen in such a way that every vertex of the set has a private neighbor (the existence of such a γset is guaranteed by Theorem 2.1.2). Statements made by Graffiti.pc are labeled accordingly, with slight abuse to the convention of labeling theorems with the last name of authors. That is, when a theorem or proposition is labelled as being due to Graffiti.pc, we mean that the statement was generated by the program. False statements retain the title of Conjecture, and true statements are labeled Proposition or Theorem. Generally, the title of Proposition is given to basic results. The title of Theorem is reserved for more difficult results and generalizations of propositions. The order in which the the conjectures are presented is not the order in which they were discussed during the project. Similar statements have been grouped together for ease of reference and (hopefully) enhanced readability. Counterexamples and special graphs are given in figures, with γ t sets indicated by unshaded vertices. 3.1 Total domination and maximum degree Lower bounds for the total domination number of a graph with conditions on the maximum degree accompany nicely the upper bounds of section 2.2, which have conditions on the minimum degree. Recall that Theorem states that, for any graph, γ t n/. If we restrict the maximum degree to be no more than a given fraction of n, then Theorem has the following corollary, which was inspired by several early conjectures generated by Graffiti.pc. Corollary Let G be a graph of order n. If n k for some positive integer k, then γ t k. Furthermore, if < n k, then γ t k + 1. Proof. By Theorem 2.1.4, γ t n/. If n/k, then substitution yields γ t k. Moreover, if < n/k, then by substitution again, we have γ t > k. Hence γ t k
17 Among the simplest conjectures made by Graffiti.pc during this study was that if < n/2, then γ t 3, which follows immediately from Corollary The program did offer an improvement to Corollary whenever n/3. Proposition (Graffiti.pc). Let G be a graph of order n. If n 3, then γ t 4. Proof. It follows immediately from Corollary that γ t 3. Suppose, by way of contradiction, that γ t = 3. Then the subgraph induced by S must be either P 3 or K 3. In either case there exists a v S adjacent to two other vertices in S, so that v can have at most 2 neighbors in V S. The other two vertices of S may have at most 1 neighbors in V S. Thus, Now, V S 2( 1) + ( 2) = 3 4. V S = n γ t = n 3. Hence, by transitivity, n Rearranging gives (n + 1)/3. This contradicts the assumption that n/3. We conclude, therefore, that γ t 4. The key idea in the proof of Proposition is that one of the vertices in S must be adjacent to at least two other vertices in S. This is the case whenever total domination number is odd. The property that the subgraph induced by S can have no isolated vertices guarantees that whenever S is odd, G[S] must have a component of order at least 3. Thus, by analogous argument to Proposition 3.1.2, we have the following improvement of Corollary for odd k. Theorem Let G be a graph of order n and let k be an odd positive integer. If n k, then γ t k + 1. Corollary and Theorem are sharp. Let k 2 be an even integer and let A k be the graph obtained as follows: begin with a cycle of order 3k and label the vertices sequentially from 1 to 3k. For each i {0, 1,..., k 2 1}, let u i and v i be the vertices labelled by 2 + 3i and 3k 3i 1, respectively, and add the edge u i v i to the cycle (see Figure 3.1 for A 4 ). Then n(a k ) = 3k and (A k ) = 3 = n k. Taking the endpoints of each of the k/2 edges that were added produces a γ t set of order k. Thus A k satisfies equality for the first inequality of Corollary for every k. For sharpness of the second inequality of Corollary 3.1.1, let k 2 be a positive integer and let B k be the graph obtained in a similar manner to A k : begin with a cycle of order 3k + 1 vertices and again label them sequentially from 1 to 3k + 1. For each i {0, 1,..., k 2 1}, let u i and v i be the vertices labeled 2 + 3i and 3k 3i, respectively, and add the edge u i v i to the cycle (see Figure 3.1 for B 3 ). Then n(b k ) = 3k + 1 and (B k ) = 3 < n k. Taking the endpoints of each edge added, and if k is odd also taking the vertex labelled 12
18 k/2, produces a γ t set of order k +1. Thus B k satisfies equality for the second bound of Corollary Finally, for sharpness of Theorem 3.1.3, let k 3 be an odd integer and let D k be the graph obtained as follows: begin with a cycle of order 3k, with the vertices labelled sequentially from 1 to 3k. For each i {0, 1,..., k 2 1}, let u i and v i be the vertices labelled 2 + 3i and 3k 3i 1, respectively, and add the edge u i v i to the cycle. Additionally, add an edge between the vertices labelled 3k 2 and 3k (see Figure 3.1 for D 3). Then n(d k ) = 3k and (D k ) = 3 = n k. Taking the endpoints of each edge added produces a γ tset of order k + 1, and equality holds for Theorem (a) A 4 (b) B 3 (c) D 3 Figure 3.1: Sharpness of Corollary and Theorem Graffiti.pc made three conjectures that involve reducing the order of the graph modulo some constant. As we will see in the following discussion, two of these conjectures are false, although the idea behind the proposed bounds is interesting. Conjecture (Graffiti.pc). Let G be graph of order n. If n 4, then γ t 1 + (n mod 5). False. We already know that γ t 4 by Corollary However, Conjecture claims that γ t 5 whenever n 4 (mod 5). Fig. 3.2 presents a 24vertex counterexample. This is a smallest counterexample, assuming the graph is connected. To see this, we need only check the truth of the conjecture for graphs on 9, 14, and 19 vertices. Let G be a 9vertex graph. We will assume for contradiction that γ t 4. Since 9/4 = 2, each vertex in a γ t set S can have at most one neighbor in V S. Thus V S γ t. Since V S = n γ t and γ t = 4, we have that 13
19 n 2γ t 8. This contradicts the assumption that n = 9, so we must conclude that γ t 5. The argument for n = 14 and n = 19 is entirely analogous. We note that a smallest disconnected counterexample is the graph formed by two disjoint copies of P 2. Figure 3.2: A smallest counterexample to Conjecture with order 24. Conjecture appeared on another list of Graffiti.pc conjectures with the added condition that δ 2. The graph in Figure 3.2 can be made into a counterexample for the modified conjecture by joining pairs of degree one vertices. This is also a smallest connected counterexample, for the same reasons as before. A smallest disconnected counterexample is the graph formed by two copies of K 3. Conjecture (Graffiti.pc). Let G be a graph of order n 8. If n 4, then γ t (n mod 4) + 3. False. Let r = n mod 4. If r 2, the statement follows from Corollary Suppose r = 3. Then Conjecture claims that γ t 6. A 19vertex counterexample is presented in Fig Observe that γ t = 5, but (n mod 4) + 3 = 6. This is a smallest counterexample. We show this by checking the cases when n = 11 and n = 15. Let G be an 11vertex graph. Then 11/4 = 2. The only connected graphs with = 2 are cycles and paths. By Proposition 1.7.2, we have γ t = (11 + 1)/2 = 6, so the conjecture is true when n = 11. Suppose n = 15. Then 15/4 = 3. By way of contradiction, suppose γ t = 5 and let S be a minimum γ t set. Then one of the vertices v S must be adjacent to at least two other vertices in S. In other words, every vertex in S can have at most two neighbors in V S, except for v, which can have at most one neighbor in V S. Thus N[S] can be at most 3γ t 1 = 14, from which we conclude that S is not a γ t set. Therefore, by way of contradiction, the conjecture holds when n = 15. We have shown that a counterexample can have order no less than 16. Since 19 is the smallest integer greater than 16 that is congruent to 3 modulo 4, the graph in Fig. 3.3 is a smallest counterexample. Graffiti.pc proposed the following true statement. Note the similarity of the statement of Proposition to the statements of the false conjectures discussed above. Proposition (Graffiti.pc). Let G be a graph of order n. If n 4, then γ t (n mod 2)
20 Figure 3.3: A 19vertex smallest counterexample to Conjecture By Corollary 3.1.1, if n/4, then γ t 4, and γ t 5 whenever < n/4. When n is not divisible by 4, then < n/4. Thus Proposition follows from Corollary However, Corollary is stronger. Notice that when n 2 (mod 4), Corollary gives γ t 5, whereas Proposition gives γ t 4. Recall that Proposition states that if n/3, then γ t 4. Since n mod 1 = 0 for every n, we may instead rewrite the right hand side of Proposition as γ t (n mod 1) + 4. Upon comparing this to Proposition 3.1.6, we proposed the following: Proposition Let G be a graph of order n. If n/5, then γ t (n mod 3) + 4. Proposition follows, in all cases, from Corollary We generalize these observations in the following result. Theorem Let G be a graph of order n. If n k integer k 3, then γ t (n mod (k 2)) + 4. for some positive Proof. Let r = n mod (k 2). As in Proposition 3.1.7, the theorem follows immediately from Corollary when r k 4. The only case left to consider is when r = k 3, for which the right hand side gives γ t k + 1. If k is odd, then the result follows from Theorem Suppose that k is even and let a be an integer such that n = a(k 2) + r. Since k is even, a(k 2) is even and r = k 3 is odd. Thus n is odd and therefore not divisible by k. This implies that < n/k, from which the theorem follows again from Corollary Observe that the value of the bound in Theorem is no more than k + 1. The bound in Corollary is therefore never worse than the bound in Theorem The idea of using the modulus operator to improve on the result in Corollary still warrants further investigation. It is not clear, however, than reducing n modulo a constant is the best approach. Finally, we mention another conjecture of Graffiti.pc involving a condition on the maximum degree, that does not resemble the preceding conjectures. Proposition (Graffiti.pc). Let G be a graph of order n. If n 3, then γ t γ. 15
21 Proof. By Proposition 3.1.2, γ t 4, so the statement is true whenever γ 4. We now show by induction on γ that if γ 4, then γ γ. If γ = 4, then γ = = 4 = γ. This establishes the base case. Suppose that the statement is true whenever γ k for some integer k 4 and let γ = k + 1. Then, by inductive hypothesis, k k = k. Using properties of the ceiling function, we have k k = (k + 1). Therefore, k (k + 1), and by the principle of mathematical induction, γ γ whenever γ 4. But γ t γ, so the proposition follows by transitivity. Observe that Proposition is never sharp whenever γ 6, since then the right hand side of the bound is strictly less than γ. However, the bound is sharp for 2 γ 5. Figure 3.4 shows a graph that is sharp for each value of γ between two and five. Each graph can be extended to an infinite family by adding any number of pendants to the support vertices. 3.2 Total domination and cut vertices A cut vertex of a graph G is any vertex c V for which G[V {c}] has more components than G. Let C be the set of cut vertices of G. Suppose G is a graph with at least one cut vertex c and let G = (V, E ) be the component of G that contains c. Denote the components of G [V {c}] by G 1, G 2,..., G k and observe that each component G i contains at least one neighbor of c. For each component G i, choose exactly one neighbor of c in the component and call this set N s (c). We will refer to this set as a set of sensitive neighbors of c. Observe that every c C has at least two sensitive neighbors, and that for any u, v N s (c) every uv path contains c. Furthermore, note that any set of sensitive neighbors is an independent set. The goal of this section is to prove the following: Theorem (Graffiti.pc). Let G be a graph. Then γ t (G) 1+ C µ(g[c]), where C is the set of cut vertices of G. Moreover, if µ(g[c]) is even and µ(g[c]) = C 2, then γ t(g) 2 + µ(g[c]). Before presenting the proof, we prove two lemmas concerning sensitive neighbors of a cut vertex and the components of the subgraph induced by a dominating set. Lemma Let G be a graph with a nonempty set of cut vertices C and let D be a dominating set of V. For any cut vertex c C D, no two sensitive neighbors of c can be in the closed neighborhood of a component of the subgraph induced by D. 16
22 (a) γ = γ t = 2 (b) γ = 3, γ t = 4 (c) γ = γ t = 4 (d) γ = 5, γ t = 5 Figure 3.4: Sharpness of Proposition
23 Proof. By way of contradiction, suppose c has two distinct sensitive neighbors u, v N s (c) that are in the closed neighborhood of some component D i of the subgraph induced by D. Since u, v N[D i ], there exist x, y D i (with possibly x = y) such that u x and v y. Observe that there exists a path from x to y containing only vertices of D i and note that this path does not contain c. Thus, since u x and v y, there exists a path from u to v that also does not contain c. This contradicts the choice of u and v as sensitive neighbors of c. Lemma Let G be a graph with a nonempty set of cut vertices C and let D be a dominating set of V (G). Let c 1, c 2 C D. If x 1, y 1 are sensitive neighbors of c 1 and x 2, y 2 are sensitive neighbors of c 2 such that x 1, x 2 N[D 1 ] and y 1, y 2 N[D 2 ], for two distinct components D 1, D 2 of G[D], then c 1 = c 2. Proof. Suppose, for contradiction, that c 1 c 2. Since x 1, x 2 N[D 1 ], there exists a path from x 1 to x 2 not involving c 1, through some vertices of D 1 ; call that path P 1. Similarly, there exists a path from y 1 to y 2, also without c 1, through some vertices of D 2 ; call that path P 2. Since x 2, y 2 N s (c 2 ), observe that c 2 is adjacent to an endpoint of each of the paths P 1 and P 2. Thus, there is an x 1 y 1 path not involving c 1, contradicting the assumption that x 1, y 1 N s (c 1 ). Finally, we mention the following basic result for simple graphs: Proposition Let G be a simple graph of order n. If E(G) n then G has at least one cycle. We are now ready to prove Theorem Proof of Theorem Let G be a graph. Proceeding by way of contradiction, suppose that γ t (G) C µ(g[c]). Let S V (G) be a γ t set with components S 1, S 2,..., S k of G[S], and let C C be the set of cut vertices in S. We examine two cases: when µ(g[c]) k, and µ(g[c]) k + 1. First, suppose µ(g[c]) k. Each component S i of G[S] has at least one edge, and therefore taking one edge from each component produces a matching M with k edges. Since µ(g[c]) k, at most µ(g[c]) edges of M may be incident to two cut vertices. Thus there are at least k µ(g[c]) edges of M for which at least one endpoint must not be a cut vertex, so γ t (G) C + k µ(g[c]). But γ t (G) C µ(g[c]), from which it follows that C µ(g[c]) C + k µ(g[c]). Adding µ(g[c]) to both sides and rearranging gives C C k. Now, C C is the number of cut vertices in V S, so there are at least as many cut vertices in V S as there are components of G[S]. On the other hand, if µ(g[c]) k + 1, then γ t (G) C µ(g[c]) C (k + 1). 18
24 Since C γ t (G), we have C C (k + 1). Rearranging gives C C k+1. In this case there are more cut vertices in V S than there are components of G[S]. Let H be the graph whose vertices are the closed neighborhoods of the components S i of G[S], labelled by their index. Note that n(h) = k. We define the edge set of H as follows: for any i, j V (H), the edge ij E(H) if there exists a cut vertex c V S with sensitive neighbors in N[S i ] and N[S j ]. By Lemmas and 3.2.3, H may not contain loops or multiedges, and is therefore simple. We showed above that the number of cut vertices in V S is at least k, so E(H) k = n(h). It follows from Proposition that H contains a cycle. Notice that for any ij path in H, there exists an xy path in G for any x N[S i ] and y N[S j ]. Let ij E(H) be an edge involved in a cycle, and let c C be the corresponding cut vertex in V S with sensitive neighbors x N[S i ] and y N[S j ]. Since ij is in a cycle, there exists an ij path in H not involving the edge ij. But this means that there exists an xy path in G not involving c, which contradicts the assumption that x, y N s (c). We conclude, therefore, that γ t (G) 1 + C µ(g[c]). Finally, we show that if µ(g[c]) is even and µ(g[c]) = C 2, then γ t(g) 2+µ(G[C]). First, we note that the statement is trivially true if µ(g[c]) = 0, so we can assume that µ(g[c]) 2. Again, let S be a γ t set and suppose by way of contradiction that γ t (G) 1 + µ(g[c]) = 1 + C 2. Then C C (1 + C 2 ) = C 2 1, where again C = C S. Note that 1 + µ(g[c]) is odd, so any γ t set can have at most C 4 components. Let H be defined the same way as above. Then n(h) C 4 and m(h) C 1. Furthermore, since µ(g[c]) 2, then C 4, which implies that 2 C 4 C 1. Since 4 = C 2 C C 4, we have 2 1 C 4. Therefore, by transitivity m(h) n(h). Thus H must contain a cycle, and we can deduce the same contradiction as before. The following corollary demonstrates the significance of the second claim in Theorem Corollary Let G be a graph. Then γ t 1 + µ(g[c]), where C is the set of cut vertices of G. Proof. Since µ(g[c]) C 2, and from Theorem 3.2.1, it follows that γ t 1 + C µ(g[c]) 1 + 2µ(G[C]) µ(g[c]) = 1 + µ(g[c]). Let T be a tree with at least 3 vertices. The set of cut vertices of T is precisely the set of nonleaves of T. Recall from Theorem of section 2.3 that γ t (T ) (n + 2 l)/2, where n is the order of the tree and l is the number of leaves. Since n l is the number of cut vertices of T, we can rewrite the bound of Theorem as γ t (T ) 1 + C /2, where C is the set of cut vertices. Thus, the next corollary generalizes Theorem to all graphs with no isolated vertices. 19
25 Corollary Let G be a graph. Then γ t 1 + C 2, where C is the set of cut vertices of G. Proof. Again, since µ(g[c]) C 2, and from Theorem 3.2.1, we have γ t 1 + C µ(g[c]) 1 + C C 2 = 1 + C 2. The proof of Corollary makes it clear that any tree satisfying equality for the bound in the corollary will also have equality for the first bound in Theorem (Chellali and Haynes characterized these trees in [3]). However, the first bound in Theorem is sharp for many other graphs. To see that this is the case, let H be the set of all graphs with at least one vertex of degree n 1. Then a family of graphs for which the bound is sharp is derived by taking any two graphs from H and joining them by an edge between vertices of maximum degree. 3.3 Total domination and degree two vertices Suppose G is a graph with minimum degree equal to one. Every vertex adjacent to a pendant must be in every γ t set. In this sense, total domination is intimately related to the set of neighbors of degree one vertices. It is obvious, however, that the number of pendants is itself not comparable to total domination number (consider stars, for example). The set of degree two vertices, although not quite as intertwined with the concept of total domination as pendants, is still closely related in the sense that a large number of the vertices in the closed neighborhood of the degree two vertices are in some γ t set. Like pendants, we cannot compare total domination number to the number of degree two vertices. Never the less, interesting bounds for total domination number can be found by investigating invariants involving the set of degree two vertices. For example, Lam and Wei [17] proved sufficient conditions for total domination number to be at most half of the order of a graph by considering the subgraph induced by V 2. Theorem (Lam and Wei [17]). Let G be a graph of order n. If δ 2 and every component of G[V 2 ] has size at most one, then γ t n 2. One way to examine the relationship between total domination number and the set of degree two vertices is to consider how the subgraph induced by the non degree two vertices is dominated. Graffiti.pc made three conjectures involving this subgraph: Conjecture A γ t (G) γ(g[v V 2 ]) + (G[V 2 ]) 1. Conjecture B Let p be the order of a largest component of G[V 2 ]. γ t (G) γ(g[v V 2 ]) p 2. Then Conjecture C γ t (G) γ(g[v V 2 ]). 20
26 V V 2 X V 2 G D S Figure 3.5: Illustration of the argument in Proposition Each conjecture is listed above in order of appearance during the study. It turns out that all three conjectures are true. However, Conjectures A and B were not proven until after we proved Conjecture C. Notice that all three conjectures are trivially true for any graph with no degree two vertices. We assume, therefore, that the set of degree two vertices in nonempty. Proposition (Graffiti.pc). Let G be a graph. Then γ t (G) γ(g[v V 2 ]). Proof. We show that γ(g[v V 2 ]) γ t (G) by constructing a dominating set of G[V V 2 ] of order at most γ t (G). Let S V be a γ t set of G, and let D = S (V V 2 ). Since S is a γ t set, every v V V 2 is adjacent to some u S. If every v V V 2 has a neighbor in D, then D is a dominating set and we are done. Suppose that at least one v V V 2 is only dominated by a u S D. Then u is a degree two vertex and as such has a second neighbor w somewhere in the graph. However, since S is a γ t set, it must be the case that w S, since otherwise G[S] would have an isolated vertex. Let X be the set of all v V V 2 that do not have neighbors in D. It follows from the preceding argument that every v X has a distinct neighbor u S D (this situation is depicted in Figure 3.5 with the dashed line representing an impossible edge). Let D = D X and observe that D S = γ t (G) and that D is a dominating set of G[V V 2 ]. Therefore, γ(g[v V 2 ]) D γ t (G). The bound in Proposition is sharp. Let G k be the 1corona of S k for some positive integer k 3. The subgraph induced by the non degree two vertices is the union of a P 2 with k 1 isolated vertices, so γ(g[v V 2 ]) = k and by Proposition 3.3.2, γ t (G k ) γ(g[v V 2 ]). The center vertex of the star and its non degree one neighbors, form a total dominating set of order k. Therefore, γ t (G k ) = γ(g[v V 2 ]). The general inadequacy of Proposition as an estimation of γ t (G) is apparent in the overall disregard for the degree two vertices. Conjectures A 21
27 and B offer improvements. It is simple to see that Conjecture A follows from Conjecture B whenever (G[V 2 ]) 1, since the order of a largest component of G[V 2 ] is always strictly greater than the maximum degree of G[V 2 ], which is never more than 2. When (G[V 2 ]) = 0, Conjecture A follows immediately from Proposition In light of this, we will focus only on Conjecture B. For aesthetic reasons, the ceiling function is removed from the statement of the conjecture. Proposition (Graffiti.pc). Let G be a graph and let p be the order of a largest component of G[V 2 ]. Then γ t (G) γ(g[v V 2 ]) + p 2 1. Proof. In a similar manner to the proof of Proposition 3.3.2, we will show that γ(g[v V 2 ]) γ t (G) p 2 1. We will also use the same construction, represented visually in Figure 3.5. Let P be a largest component of G[V 2 ] and let S be a γ t set of G[P ]. If P 2, then the statement follows from Proposition 3.3.2, so we can assume that P 3. Observe that P is either a cycle or a path. If P is a cycle, then S p/2 (the total domination number of paths and cycles is characterized in section 1.7). Moreover, none of the vertices of S can have neighbors in X. Hence D = D X S S = γ t (G) S γ t (G) p 2. If P is a path, observe that every vertex of P that is not an endpoint of P can not have a neighbor in X. If neither of the endpoints of P are elements of S, then the number of vertices of P that are elements of S is γ t (G[P ]) and hence S p/2. If both endpoints of P are elements of S, then instead of taking S to be a γ t set of the entire path, we take S to be a γ t set of the path excluding the endpoints of P. Then S (p 2)/2. Finally, if only one endpoint of P is an element of S, then we take S to be a γ t set of the path excluding this endpoint of P. Thus S (p 1)/2. In every case, S (p 2)/2. Therefore, D = D X S S = γ t (G) S γ t (G) p 2 2. Since γ(g[v V 2 ]) D, we have, after simplifying and rearranging, γ t (G) γ(g[v V 2 ]) + p 2 1. The bound in Proposition is sharp for the graph H k, which is formed by connecting the endpoints of P 2 to the center of the 1corona of S k (see Figure 3.6 for H 3 ; by the center, we mean the same vertex as the center of S k ). Observe that γ t (H k ) = γ(h k [V V 2 ]) = k. The order of a largest component is two, so the right hand side of the bound in Proposition is just γ(h k [V V 2 ]). Additionally, notice that Proposition follows from Proposition whenever p 3. On the other hand, Proposition follows from Proposition whenever p 2. Proposition has the following corollary. 22
28 Figure 3.6: Sharpness of Proposition Corollary Let G be a graph. Let c be the number of components of G[V V 2 ] and let p be the order of a largest component of G[V 2 ]. Then γ t (G) c + p/2 1. Proof. Every γset of G[V V 2 ] contains at least one vertex from each component of the induced subgraph. That is, γ(g[v V 2 ]) c. Therefore, by Proposition 3.3.3, γ t (G) γ(g[v V 2 ]) + p 2 1 c + p 2 1. Observe that Theorem 2.3.2, which was also a conjecture of Graffiti.pc, now follows from Corollary by simply taking G to be a nontrivial tree. Next, we show that the argument for Proposition can be extended to every component of G[V 2 ], resulting in an even stronger lower bound. Theorem Let G be a graph and let k be the number of components of G[V 2 ]. Then γ t (G) γ(g[v V 2 ]) V 2 k. Proof. Again, we make frequent use of the same construction as in the arguments for Propositions and (see Figure 3.5). Let k be the number of components of G[V 2 ]. Let P i be the ith component and denote p i = P i. Order the components by p i in nonincreasing order and let j be the largest index of a component with order at least 3. If p i 3 then we can construct a set S i S for each component so that each vertex in S i has no neighbors in X (see the proof of Proposition 3.3.3). That is, D = D X S j i=1 S i = S j S i. We showed in the proof of Proposition that S i (p i 2)/2 for every i j. Recall that D is a dominating set of G[V V 2 ]. It follows that γ(g[v V 2 ]) S j S i S i=1 i=1 j i=1 p i 2. ( ) 2 Finally, since p i 2 for j < i k, we have (p i 2)/2 0. Thus, (p i 2)/2 0, so subtracting these terms from the right hand side of ( ) maintains the inequality. Therefore, 23
29 γ(g[v V 2 ]) S k i=1 p i 2. 2 Now, observe that k i=1 p i = V 2. After rearranging the inequality, and using that S = γ t (G), we have γ t (G) γ(g[v V 2 ]) + = γ(g[v V 2 ]) + k i=1 i=1 p i 2 2 k p i k 2 1 = γ(g[v V 2 ]) V 2 k. i=1 Notice that Theorem is sharp for the same family H k for Proposition Graffiti.pc made an interesting conjecture relating total domination number to the difference between the cardinality of the open neighborhood of the set of degree two vertices and the number of degree two vertices. Theorem (Graffiti.pc). Let G be a graph. Then γ t N(V 2 ) V 2. Proof. The theorem is trivially true if N(V 2 ) V 2 2, so we can assume that N(V 2 ) V 2 k for some positive integer k 3. For every v V 2, assign to v a u N(v) in such a way that no two vertices of degree two are assigned the same vertex. Observe that there remain k vertices of N(V 2 ) that have not been assigned to any degree two vertex. Let X be the set of these k remaining vertices. Since every degree two vertex has been assigned to a distinct neighbor, no two vertices of X can have the same neighbor v V 2, since otherwise v would have degree at least three. Thus there exist k vertices of degree two that have no common neighbors. However, by definition of total domination, each of these k degree two vertices is adjacent to a vertex in some total dominating set. Thus, γ t k = N(V 2 ) V 2. Upon first glance, one may wonder why Graffiti.pc didn t conjecture instead that γ t (G) N(V 2 ) V 2. Fig. 3.7 shows a 9vertex counterexample to this statement, a graph that is definitely in the database given to Graffiti.pc. Note that N(V 2 ) V 2 = 5, but γ t = 3. Theorem is sharp, as can be seen by the following construction. Let k be a positive integer and let L k be the graph obtained as follows. Begin with k copies of K 3 and k isolated vertices. From each copy of K 3 select two vertices and add every possible edge between the chosen vertices and the k isolated vertices (see Figure 3.8 for L 2 ). Notice that N(V 2 ) V 2 = k and γ t (L k ) = k, therefore satisfying equality for Theorem
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