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
Contents
1 Graph Theory Background
1.1 The definition of a graph . . . . . . . . . . . .
1.2 Neighborhoods and degree . . . . . . . . . . .
1.3 Distance, radius, and diameter . . . . . . . .
1.4 Domination and total domination . . . . . . .
1.5 Independence and matchings . . . . . . . . .
1.6 Some special families of graphs . . . . . . . .
1.7 Characterizations of total domination number
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1
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2 A Short Survey of Known Results
2.1 Some basic results for domination and total domination
2.2 Total domination and minimum degree . . . . . . . . . .
2.3 Total domination in trees . . . . . . . . . . . . . . . . .
2.4 Total domination in planar graphs . . . . . . . . . . . .
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3 Conjectures of Graffiti.pc
3.1 Total domination and maximum degree .
3.2 Total domination and cut vertices . . . .
3.3 Total domination and degree two vertices
3.4 Total Domination and G2 . . . . . . . . . .
3.5 Miscellaneous conjectures . . . . . . . . .
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4 Concluding Remarks
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33
List of Figures
2.1
2.2
The graphs G10 (left) and H10 from Theorem 2.2.4. . . . . . . .
The unique planar graph with diameter 2 and γ = 3. . . . . . . .
8
10
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
3.12
3.13
3.14
3.15
Sharpness of Corollary 3.1.1 and Theorem 3.1.3. . . . . . . .
A smallest counterexample to Conjecture 3.1.4 with order 24.
A 19-vertex smallest counterexample to Conjecture 3.1.5. . .
Sharpness of Proposition 3.1.9. . . . . . . . . . . . . . . . . .
Illustration of the argument in Proposition 3.3.2. . . . . . . .
Sharpness of Proposition 3.3.3. . . . . . . . . . . . . . . . . .
9-vertex counterexample to γt (G) ≥ |N (V2 ) − V2 |. . . . . . . .
Sharpness of Theorem 3.3.6. . . . . . . . . . . . . . . . . . . .
The graph Ŝ5 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A 16-vertex counterexample to Conjecture 3.5.1. . . . . . . .
A 21-vertex counterexample to Conjecture 3.5.2. . . . . . . .
An 18-vertex counterexample to Conjecture 3.5.3. . . . . . . .
A 24-vertex counterexample to Conjecture 3.5.4. . . . . . . .
A 14-vertex counterexample to Conjectures 3.5.5 and 3.5.6. .
A 13-vertex counterexample to Conjecture 3.5.7. . . . . . . .
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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 1980. 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 γt (G) is the cardinality of a smallest
total dominating set. The problem of computing γt (G) for a given graph was
shown to be NP-complete by Pfaff et al. [20]. However, a linear algorithm
exists for trees [18]. The overall complexity of calculating γt (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.
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.
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 multi-edge if it appears in E more than once. A graph whose edge
set contains no loops and no multi-edges 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 {x1 , x2 , . . . , xk }
such that x1 ∼ x2 , x2 ∼ x3 , . . . , xk−1 ∼ xk . x1 and x2 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 u-v path. A cycle is a path with
the added condition that xk ∼ x1 . 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 0 = {uv ∈ E(G) : u, v ∈ X}.
1
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 dG (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 Vk = {v ∈ V : d(v) = k}. Hence Vδ is the set of
minimum degree vertices, and V2 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 u-v 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 γ-set.
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 γt = γt (G), is the cardinality of a smallest total dominating set, and
we refer to such a set as a γt -set. An immediate consequence of the definitions
of domination number and total domination number is that, for any graph G,
γt ≥ γ.
2
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 1.5.1. 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 1.5.2. 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 Pn , and is the graph with V (Pn ) = {v1 , v2 , . . . , vn }
and E(Pn ) = {v1 v2 , v2 v3 , . . . , vn−1 vn }. The cycle on n vertices is denoted Cn
and has V (Cn ) = V (Pn ) and E(Cn ) = E(Pn ) ∪ {vn v1 }.
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 Kn . 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
Kn1 ,n2 where n1 = |A| and n2 = |B|.
The star on n vertices, denoted Sn is, is equivalent to K1,n−1 . The degree
n − 1 vertex of Sn is called the center. A binary star is made up of two stars,
with an edge added between the centers.
3
The k-corona of a connected graph H, denoted H ◦Pk , is a the graph formed
as follows: Take n(H) copies of Pk and let X be the set made up of exactly one
endpoint from each copy of Pk . Let f : V (H) → X be a bijection. Connect
each v ∈ V (H) to an endpoint of a corresponding Pk 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 1.7.1. Let G be a graph of order n with no isolated vertices. If
∆ ≥ n − 2, then γt = 2.
Graphs for which Observation 1.7.1 applies include stars and complete graphs.
Other graphs with γt = 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
Proposition 1.7.2. The total domination number of a cycle Cn or a path Pn
on n ≥ 3 vertices is given by:
(a) γt (Cn ) = γt (Pn ) =
n
2
(b) γt (Cn ) = γt (Pn ) =
n+2
2
if n ≡ 2(mod 4).
(c) γt (Cn ) = γt (Pn ) =
n+1
2
otherwise.
if n ≡ 0(mod 4).
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
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 2.1.1. Let G be a graph with no isolated vertices. Then γ ≤
n
2.
Proof. Let D ⊆ V (G) be a γ-set. 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 γ-set. Thus γ = |D| ≤ n/2.
Bollobás and Cockayne proved a useful property of γ-sets in [1].
Theorem 2.1.2 (Bollobás and Cockayne [1]). Every graph G with no isolated
vertices has a γ-set D such that every v ∈ D has the property that there exists
a vertex v 0 ∈ V − D that is adjacent to v but to no other vertex of D.
The vertex v 0 from Theorem 2.1.2 is called a private neighbor of v. From
now on, we will assume that every γ-set is chosen to be one whose existence
is guaranteed by Theorem 2.1.2. This allows us to prove the following triple
inequality.
Theorem 2.1.3. Let G be a graph with no isolated vertices. Then γ ≤ γt ≤ 2γ.
6
Proof. The first inequality follows immediately from the definitions of domination number and total domination number. Let D be a γ-set. 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 D0 . At most |D| private neighbors
could have been added to form D0 . That is, |D0 | ≤ 2|D|. Since D0 is a total
dominating set, |D0 | ≥ γt . Therefore, γt ≤ 2|D| = 2γ. This proves the second
inequality.
Finally, we mention a basic lower bound for total domination number.
Theorem 2.1.4. Let G be a graph of order n with no isolated vertices. Then
n
γt ≥ ∆
.
Proof. Let S be a γt -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 ∆γt ≥ |V | = n. The theorem follows by dividing
this inequality by ∆.
In section 3.1, we prove a few corollaries to Theorem 2.1.4.
2.2
Total domination and minimum degree
Cockayne et al. [6] proved the following for connected graphs.
Theorem 2.2.1 (Cockayne et al. [6]). Let G be a connected graph of order
n ≥ 3 with no isolated vertices. Then γt ≤ 32 n.
Equality for Theorem 2.2.1 was characterized by Brigham, Carrington, and
Vitray [2].
Theorem 2.2.2 (Brigham, Carrington, and Vitray [2]). Let G be a connected
graph of order n ≥ 3. Then γt = 2n/3 if and only if G is C3 , C6 or H ◦ P2 for
some connected graph H.
Sun [21] showed that the bound in Theorem 2.2.1 can be improved upon for
connected graphs with minimum degree at least two.
Theorem 2.2.3 (Sun [21]). Let G be a connected graph of order n and δ ≥ 2.
Then γt ≤ b 47 (n + 1)c.
Henning [14] improved this bound slightly by disregarding six graphs with
small orders. The graphs G10 and H10 are displayed in Fig. 2.1.
Theorem 2.2.4 (Henning [14]). If G 6∈ {C3 , C5 , C6 , C10 , G10 , H10 } is a connected graph of order n and δ ≥ 2, then γt ≤ 74 n.
Chvátal and McDiarmid [4] and Tuza [24] independently proved a theorem
concerning transversals in hypergraphs. Theorem 2.2.5 is a consequence of their
result.
7
Figure 2.1: The graphs G10 (left) and H10 from Theorem 2.2.4.
Theorem 2.2.5 (Chvátal and McDiarmid [4]; Tuza [24]). Let G be a graph of
order n and δ ≥ 3. Then γt ≤ n2 .
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 2.2.6 (Thomassé and Yeo [22]). Let G be a graph of order n and
δ ≥ 4. Then γt ≤ 37 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 γt -set or are in no γt set [5]. For example, support vertices must be in every γt -set. Furthermore,
Haynes and Henning found three conditions that are equivalent to a tree having
a unique minimum γt -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 2.3.1 (Chellali and Haynes [3]). Let T be a nontrivial tree of order
n and l leaves. Then γt ≥ n+2−l
.
2
8
We show in section 3.2 that Theorem 2.3.1 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 2.3.2 (DeLaViña et al. [7]). Let T be a non-trivial tree. Let c2 be
the number of components of the subgraph induced by V (T ) − V2 , and let p2 be
the order of a largest path in the subgraph induced by V2 . Then γt ≥ c2 + p22 − 1.
Theorem 2.3.2 is also a corollary of one of Graffiti.pc’s conjectures, as will
be seen in section 3.3.
2.4
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 1-corona 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 2.4.1 (MacGillivray and Seyffarth [19]). If G is a planar graph with
diameter two, then γ ≤ 3.
Theorem 2.4.2 (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 γt ≤ 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 2.4.3 (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 γt ≤ 3.
(b) If diam(G) = 3 and rad(G) = 2, then γt ≤ 5.
(c) If diam(G) = 3 and rad(G) = 3, then γt ≤ 10.
(d) If diam(G) = 3 and G has sufficiently large order, then γt ≤ 7.
9
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
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 γ-set is chosen in
such a way that every vertex of the set has a private neighbor (the existence of
such a γ-set 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 γt -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 2.1.4 states that,
for any graph, γt ≥ n/∆. If we restrict the maximum degree to be no more than
a given fraction of n, then Theorem 2.1.4 has the following corollary, which was
inspired by several early conjectures generated by Graffiti.pc.
Corollary 3.1.1. Let G be a graph of order n. If ∆ ≤ nk for some positive
integer k, then γt ≥ k. Furthermore, if ∆ < nk , then γt ≥ k + 1.
Proof. By Theorem 2.1.4, γt ≥ n/∆. If ∆ ≤ n/k, then substitution yields
γt ≥ k. Moreover, if ∆ < n/k, then by substitution again, we have γt > k.
Hence γt ≥ k + 1.
11
Among the simplest conjectures made by Graffiti.pc during this study was
that if ∆ < n/2, then γt ≥ 3, which follows immediately from Corollary 3.1.1.
The program did offer an improvement to Corollary 3.1.1 whenever ∆ ≤ n/3.
Proposition 3.1.2 (Graffiti.pc). Let G be a graph of order n. If ∆ ≤
γt ≥ 4.
n
3,
then
Proof. It follows immediately from Corollary 3.1.1 that γt ≥ 3. Suppose, by way
of contradiction, that γt = 3. Then the subgraph induced by S must be either
P3 or K3 . 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,
|V − S| ≤ 2(∆ − 1) + (∆ − 2) = 3∆ − 4.
Now,
|V − S| = n − γt = n − 3.
Hence, by transitivity, n − 3 ≤ 3∆ − 4. Rearranging gives ∆ ≥ (n + 1)/3.
This contradicts the assumption that ∆ ≤ n/3. We conclude, therefore, that
γt ≥ 4.
The key idea in the proof of Proposition 3.1.2 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 3.1.1 for
odd k.
Theorem 3.1.3. Let G be a graph of order n and let k be an odd positive
integer. If ∆ ≤ nk , then γt ≥ k + 1.
Corollary 3.1.1 and Theorem 3.1.3 are sharp. Let k ≥ 2 be an even integer
and let Ak 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, . . . , k2 − 1}, let
ui and vi be the vertices labelled by 2 + 3i and 3k − 3i − 1, respectively, and
add the edge ui vi to the cycle (see Figure 3.1 for A4 ). Then n(Ak ) = 3k and
∆(Ak ) = 3 = nk . Taking the endpoints of each of the k/2 edges that were added
produces a γt -set of order k. Thus Ak satisfies equality for the first inequality
of Corollary 3.1.1 for every k.
For sharpness of the second inequality of Corollary 3.1.1, let k ≥ 2 be a
positive integer and let Bk be the graph obtained in a similar manner to Ak :
begin with a cycle of order 3k + 1 vertices and again label them sequentially
from 1 to 3k + 1. For each i ∈ {0, 1, . . . , b k2 c − 1}, let ui and vi be the vertices
labeled 2 + 3i and 3k − 3i, respectively, and add the edge ui vi to the cycle (see
Figure 3.1 for B3 ). Then n(Bk ) = 3k + 1 and ∆(Bk ) = 3 < nk . Taking the
endpoints of each edge added, and if k is odd also taking the vertex labelled
12
bk/2c, produces a γt -set of order k + 1. Thus Bk satisfies equality for the second
bound of Corollary 3.1.1.
Finally, for sharpness of Theorem 3.1.3, let k ≥ 3 be an odd integer and
let Dk 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, . . . , b k2 c − 1},
let ui and vi be the vertices labelled 2 + 3i and 3k − 3i − 1, respectively, and
add the edge ui vi to the cycle. Additionally, add an edge between the vertices
3k
labelled b 3k
2 c and b 2 + 2c (see Figure 3.1 for D3 ). Then n(Dk ) = 3k and
n
∆(Dk ) = 3 = k . Taking the endpoints of each edge added produces a γt -set of
order k + 1, and equality holds for Theorem 3.1.3.
(a) A4
(b) B3
(c) D3
Figure 3.1: Sharpness of Corollary 3.1.1 and Theorem 3.1.3.
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 3.1.4 (Graffiti.pc). Let G be graph of order n. If ∆ ≤
γt ≥ 1 + (n mod 5).
n
4,
then
False. We already know that γt ≥ 4 by Corollary 3.1.1. However, Conjecture 3.1.4 claims that γt ≥ 5 whenever n ≡ 4 (mod 5). Fig. 3.2 presents
a 24-vertex 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 9-vertex graph. We will assume for contradiction that γt ≤ 4.
Since ∆ ≤ b9/4c = 2, each vertex in a γt -set S can have at most one neighbor
in V − S. Thus |V − S| ≤ γt . Since |V − S| = n − γt and γt = 4, we have that
13
n ≤ 2γt ≤ 8. This contradicts the assumption that n = 9, so we must conclude
that γt ≥ 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 P2 .
Figure 3.2: A smallest counterexample to Conjecture 3.1.4 with order 24.
Conjecture 3.1.4 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 K3 .
Conjecture 3.1.5 (Graffiti.pc). Let G be a graph of order n ≥ 8. If ∆ ≤
then γt ≥ (n mod 4) + 3.
n
4,
False. Let r = n mod 4. If r ≤ 2, the statement follows from Corollary
3.1.1. Suppose r = 3. Then Conjecture 3.1.5 claims that γt ≥ 6. A 19-vertex
counterexample is presented in Fig. 3.3. Observe that γt = 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 11-vertex graph. Then ∆ ≤ b11/4c = 2. The only connected
graphs with ∆ = 2 are cycles and paths. By Proposition 1.7.2, we have γt =
(11 + 1)/2 = 6, so the conjecture is true when n = 11. Suppose n = 15. Then
∆ ≤ b15/4c = 3. By way of contradiction, suppose γt = 5 and let S be a
minimum γt -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 3γt − 1 = 14, from which we conclude that S is not
a γt -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 3.1.6 to the statements of the false conjectures
discussed above.
Proposition 3.1.6 (Graffiti.pc). Let G be a graph of order n. If ∆ ≤
γt ≥ (n mod 2) + 4.
14
n
4,
then
Figure 3.3: A 19-vertex smallest counterexample to Conjecture 3.1.5.
By Corollary 3.1.1, if ∆ ≤ n/4, then γt ≥ 4, and γt ≥ 5 whenever ∆ < n/4.
When n is not divisible by 4, then ∆ < n/4. Thus Proposition 3.1.6 follows
from Corollary 3.1.1. However, Corollary 3.1.1 is stronger. Notice that when
n ≡ 2 (mod 4), Corollary 3.1.1 gives γt ≥ 5, whereas Proposition 3.1.6 gives
γt ≥ 4.
Recall that Proposition 3.1.2 states that if ∆ ≤ n/3, then γt ≥ 4. Since n
mod 1 = 0 for every n, we may instead rewrite the right hand side of Proposition 3.1.2 as γt ≥ (n mod 1) + 4. Upon comparing this to Proposition 3.1.6, we
proposed the following:
Proposition 3.1.7. Let G be a graph of order n. If ∆ ≤ n/5, then γt ≥
(n mod 3) + 4.
Proposition 3.1.7 follows, in all cases, from Corollary 3.1.1. We generalize
these observations in the following result.
Theorem 3.1.8. Let G be a graph of order n. If ∆ ≤
integer k ≥ 3, then γt ≥ (n mod (k − 2)) + 4.
n
k
for some positive
Proof. Let r = n mod (k − 2). As in Proposition 3.1.7, the theorem follows
immediately from Corollary 3.1.1 when r ≤ k − 4. The only case left to consider
is when r = k − 3, for which the right hand side gives γt ≥ k + 1. If k is odd,
then the result follows from Theorem 3.1.3. 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 3.1.1.
Observe that the value of the bound in Theorem 3.1.8 is no more than
k + 1. The bound in Corollary 3.1.1 is therefore never worse than the bound in
Theorem 3.1.8. The idea of using the modulus operator to improve on the result
in Corollary 3.1.1 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 3.1.9 (Graffiti.pc). Let G be a graph of order n. If ∆ ≤
γt ≥ 1 + d1 + 12 γe.
15
n
3,
then
Proof. By Proposition 3.1.2, γt ≥ 4, so the statement is true whenever γ ≤ 4.
We now show by induction on γ that if γ ≥ 4, then γ ≥ 1 + d1 + 12 γe. If γ = 4,
then 1 + d1 + 12 γe = 1 + d3e = 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,
l1m l
l
1 m
1 m
+ 1+ k .
k+1≥2+ 1+ k =1+
2
2
2
Using properties of the ceiling function, we have
l1m l
l1
l
m
1 m
1 m
1
+ 1+ k ≥1+
1+
+ 1 + k = 1 + 1 + (k + 1) .
2
2
2
2
2
Therefore, k + 1 ≥ 1 + d1 + 12 (k + 1)e, and by the principle of mathematical
induction, γ ≥ 1 + d1 + 12 γe whenever γ ≥ 4. But γt ≥ γ, so the proposition
follows by transitivity.
Observe that Proposition 3.1.9 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 G0 = (V 0 , E 0 ) be the component of
G that contains c. Denote the components of G0 [V 0 − {c}] by G1 , G2 , . . . , Gk
and observe that each component Gi contains at least one neighbor of c. For
each component Gi , choose exactly one neighbor of c in the component and
call this set Ns (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 ∈ Ns (c) every u-v 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 3.2.1 (Graffiti.pc). Let G be a graph. Then γt (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 γt (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 3.2.2. 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
(a) γ = γt = 2
(b) γ = 3, γt = 4
(c) γ = γt = 4
(d) γ = 5, γt = 5
Figure 3.4: Sharpness of Proposition 3.1.9.
17
Proof. By way of contradiction, suppose c has two distinct sensitive neighbors
u, v ∈ Ns (c) that are in the closed neighborhood of some component Di of the
subgraph induced by D. Since u, v ∈ N [Di ], there exist x, y ∈ Di (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 Di 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 3.2.3. Let G be a graph with a nonempty set of cut vertices C and
let D be a dominating set of V (G). Let c1 , c2 ∈ C − D. If x1 , y1 are sensitive
neighbors of c1 and x2 , y2 are sensitive neighbors of c2 such that x1 , x2 ∈ N [D1 ]
and y1 , y2 ∈ N [D2 ], for two distinct components D1 , D2 of G[D], then c1 = c2 .
Proof. Suppose, for contradiction, that c1 6= c2 . Since x1 , x2 ∈ N [D1 ], there
exists a path from x1 to x2 not involving c1 , through some vertices of D1 ;
call that path P1 . Similarly, there exists a path from y1 to y2 , also without
c1 , through some vertices of D2 ; call that path P2 . Since x2 , y2 ∈ Ns (c2 ),
observe that c2 is adjacent to an endpoint of each of the paths P1 and P2 .
Thus, there is an x1 -y1 path not involving c1 , contradicting the assumption
that x1 , y1 ∈ Ns (c1 ).
Finally, we mention the following basic result for simple graphs:
Proposition 3.2.4. 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 3.2.1.
Proof of Theorem 3.2.1. Let G be a graph. Proceeding by way of contradiction,
suppose that γt (G) ≤ |C|−µ(G[C]). Let S ⊆ V (G) be a γt -set with components
S1 , S2 , . . . , Sk of G[S], and let C 0 ⊆ 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 Si 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 γt (G) ≥ |C 0 | + k −
µ(G[C]). But γt (G) ≤ |C| − µ(G[C]), from which it follows that
|C| − µ(G[C]) ≥ |C 0 | + k − µ(G[C]).
Adding µ(G[C]) to both sides and rearranging gives |C| − |C 0 | ≥ k. Now,
|C| − |C 0 | 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
γt (G) ≤ |C| − µ(G[C]) ≤ |C| − (k + 1).
18
Since |C 0 | ≤ γt (G), we have |C 0 | ≤ |C| − (k + 1). Rearranging gives |C| − |C 0 | ≥
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 Si 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 [Si ] and N [Sj ].
By Lemmas 3.2.2 and 3.2.3, H may not contain loops or multi-edges, 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 3.2.4 that H
contains a cycle. Notice that for any i-j path in H, there exists an x-y path in
G for any x ∈ N [Si ] and y ∈ N [Sj ].
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 [Si ] and y ∈ N [Sj ].
Since ij is in a cycle, there exists an i-j path in H not involving the edge
ij. But this means that there exists an x-y path in G not involving c, which
contradicts the assumption that x, y ∈ Ns (c). We conclude, therefore, that
γt (G) ≥ 1 + |C| − µ(G[C]).
Finally, we show that if µ(G[C]) is even and µ(G[C]) = |C|
2 , then γt (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 γt -set and suppose by way of
|C|
0
contradiction that γt (G) ≤ 1 + µ(G[C]) = 1 + |C|
2 . Then |C | ≥ |C| − (1 + 2 ) =
|C|
0
2 − 1, where again C = C − S. Note that 1 + µ(G[C]) is odd, so any γt -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|
|C|
|C|
|C|
|C|
|C|
4 ≥ 1. Since 4 = 2 − 4 , we have 2 − 1 ≥ 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 3.2.1.
Corollary 3.2.5. Let G be a graph. Then γt ≥ 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
γt ≥ 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 non-leaves of T . Recall from Theorem 2.3.1 of section 2.3
that γt (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 2.3.1 as γt (T ) ≥ 1 + |C|/2, where C is the set of cut vertices.
Thus, the next corollary generalizes Theorem 2.3.1 to all graphs with no isolated
vertices.
19
Corollary 3.2.6. Let G be a graph. Then γt ≥ 1 +
cut vertices of G.
Proof. Again, since µ(G[C]) ≤
|C|
2 ,
|C|
2 ,
where C is the set of
and from Theorem 3.2.1, we have
γt ≥ 1 + |C| − µ(G[C]) ≥ 1 + |C| −
|C|
|C|
=1+
.
2
2
The proof of Corollary 3.2.6 makes it clear that any tree satisfying equality
for the bound in the corollary will also have equality for the first bound in
Theorem 3.2.1 (Chellali and Haynes characterized these trees in [3]). However,
the first bound in Theorem 3.2.1 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 γt -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 γt -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
V2 .
Theorem 3.3.1 (Lam and Wei [17]). Let G be a graph of order n. If δ ≥ 2
and every component of G[V2 ] has size at most one, then γt ≤ n2 .
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 γt (G) ≥ γ(G[V − V2 ]) + ∆(G[V2 ]) − 1.
Conjecture B Let p be the order of a largest component of G[V2 ]. Then
γt (G) ≥ γ(G[V − V2 ]) + d−1 + p2 e.
Conjecture C γt (G) ≥ γ(G[V − V2 ]).
20
V − V2
X
V2
G
S
D
Figure 3.5: Illustration of the argument in Proposition 3.3.2.
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 3.3.2 (Graffiti.pc). Let G be a graph. Then γt (G) ≥ γ(G[V −V2 ]).
Proof. We show that γ(G[V − V2 ]) ≤ γt (G) by constructing a dominating set
of G[V − V2 ] of order at most γt (G). Let S ⊆ V be a γt -set of G, and let
D = S ∩ (V − V2 ). Since S is a γt -set, every v ∈ V − V2 is adjacent to some
u ∈ S. If every v ∈ V − V2 has a neighbor in D, then D is a dominating set
and we are done. Suppose that at least one v ∈ V − V2 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 γt -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 −V2 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 D0 = D ∪ X and observe that |D0 | ≤ |S| = γt (G) and that
D0 is a dominating set of G[V −V2 ]. Therefore, γ(G[V −V2 ]) ≤ |D0 | ≤ γt (G).
The bound in Proposition 3.3.2 is sharp. Let Gk be the 1-corona of Sk for
some positive integer k ≥ 3. The subgraph induced by the non degree two
vertices is the union of a P2 with k − 1 isolated vertices, so γ(G[V − V2 ]) = k
and by Proposition 3.3.2, γt (Gk ) ≥ γ(G[V − V2 ]). The center vertex of the
star and its non degree one neighbors, form a total dominating set of order k.
Therefore, γt (Gk ) = γ(G[V − V2 ]).
The general inadequacy of Proposition 3.3.2 as an estimation of γt (G) is
apparent in the overall disregard for the degree two vertices. Conjectures A
21
and B offer improvements. It is simple to see that Conjecture A follows from
Conjecture B whenever ∆(G[V2 ]) ≥ 1, since the order of a largest component
of G[V2 ] is always strictly greater than the maximum degree of G[V2 ], which
is never more than 2. When ∆(G[V2 ]) = 0, Conjecture A follows immediately
from Proposition 3.3.2. 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 3.3.3 (Graffiti.pc). Let G be a graph and let p be the order of a
largest component of G[V2 ]. Then γt (G) ≥ γ(G[V − V2 ]) + p2 − 1.
Proof. In a similar manner to the proof of Proposition 3.3.2, we will show that
γ(G[V − V2 ]) ≤ γt (G) − d p2 − 1e. We will also use the same construction,
represented visually in Figure 3.5. Let P be a largest component of G[V2 ] and
let S 0 be a γt -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 0 | ≥ p/2 (the total domination number of paths and cycles
is characterized in section 1.7). Moreover, none of the vertices of S 0 can have
neighbors in X. Hence
p
|D0 | = |D ∪ X| ≤ |S − S 0 | = γt (G) − |S 0 | ≤ γt (G) − .
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 γt (G[P ]) and hence
|S 0 | ≥ p/2. If both endpoints of P are elements of S, then instead of taking S 0
to be a γt -set of the entire path, we take S 0 to be a γt -set of the path excluding
the endpoints of P . Then |S 0 | ≥ (p − 2)/2. Finally, if only one endpoint of P
is an element of S, then we take S 0 to be a γt -set of the path excluding this
endpoint of P . Thus |S 0 | ≥ (p − 1)/2.
In every case, |S 0 | ≥ (p − 2)/2. Therefore,
|D0 | = |D ∪ X| ≤ |S − S 0 | = γt (G) − |S 0 | ≤ γt (G) −
p−2
.
2
Since γ(G[V − V2 ]) ≤ |D0 |, we have, after simplifying and rearranging,
p
γt (G) ≥ γ(G[V − V2 ]) + − 1.
2
The bound in Proposition 3.3.3 is sharp for the graph Hk , which is formed
by connecting the endpoints of P2 to the center of the 1-corona of Sk (see Figure
3.6 for H3 ; by the center, we mean the same vertex as the center of Sk ). Observe
that γt (Hk ) = γ(Hk [V − V2 ]) = k. The order of a largest component is two,
so the right hand side of the bound in Proposition 3.3.3 is just γ(Hk [V − V2 ]).
Additionally, notice that Proposition 3.3.2 follows from Proposition 3.3.3 whenever p ≥ 3. On the other hand, Proposition 3.3.3 follows from Proposition 3.3.2
whenever p ≤ 2.
Proposition 3.3.3 has the following corollary.
22
Figure 3.6: Sharpness of Proposition 3.3.3.
Corollary 3.3.4. Let G be a graph. Let c be the number of components of
G[V − V2 ] and let p be the order of a largest component of G[V2 ]. Then γt (G) ≥
c + p/2 − 1.
Proof. Every γ-set of G[V −V2 ] contains at least one vertex from each component
of the induced subgraph. That is, γ(G[V − V2 ]) ≥ c. Therefore, by Proposition
3.3.3,
p
p
γt (G) ≥ γ(G[V − V2 ]) + − 1 ≥ c + − 1.
2
2
Observe that Theorem 2.3.2, which was also a conjecture of Graffiti.pc, now
follows from Corollary 3.3.4 by simply taking G to be a nontrivial tree. Next,
we show that the argument for Proposition 3.3.3 can be extended to every
component of G[V2 ], resulting in an even stronger lower bound.
Theorem 3.3.5. Let G be a graph and let k be the number of components of
G[V2 ]. Then γt (G) ≥ γ(G[V − V2 ]) + 21 |V2 | − k.
Proof. Again, we make frequent use of the same construction as in the arguments for Propositions 3.3.2 and 3.3.3 (see Figure 3.5). Let k be the number of
components of G[V2 ]. Let Pi be the ith component and denote pi = |Pi |. Order
the components by pi in nonincreasing order and let j be the largest index of a
component with order at least 3. If pi ≥ 3 then we can construct a set Si0 ⊆ S
for each component so that each vertex in Si0 has no neighbors in X (see the
proof of Proposition 3.3.3). That is,
j
j
[
X
|D0 | = |D ∪ X| ≤ S −
Si0 = |S| −
|Si0 |.
i=1
i=1
|Si0 |
We showed in the proof of Proposition 3.3.3 that
≥ (pi − 2)/2 for every
i ≤ j. Recall that D0 is a dominating set of G[V − V2 ]. It follows that
γ(G[V − V2 ]) ≤ |S| −
j
X
i=1
|Si0 | ≤ |S| −
j
X
pi − 2
i=1
2
.
(∗)
Finally, since pi ≤ 2 for j < i ≤ k, we have (pi − 2)/2 ≤ 0. Thus, −(pi − 2)/2 ≥
0, so subtracting these terms from the right hand side of (∗) maintains the
inequality. Therefore,
23
γ(G[V − V2 ]) ≤ |S| −
k
X
pi − 2
2
i=1
.
Pk
Now, observe that i=1 pi = |V2 |. After rearranging the inequality, and using
that |S| = γt (G), we have
γt (G) ≥ γ(G[V − V2 ]) +
k
X
pi − 2
2
i=1
= γ(G[V − V2 ]) +
k
X
pi
i=1
2
−
k
X
1
i=1
1
= γ(G[V − V2 ]) + |V2 | − k.
2
Notice that Theorem 3.3.5 is sharp for the same family Hk for Proposition
3.3.3. 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 3.3.6 (Graffiti.pc). Let G be a graph. Then γt ≥ |N (V2 )| − |V2 |.
Proof. The theorem is trivially true if |N (V2 )| − |V2 | ≤ 2, so we can assume that
|N (V2 )| − |V2 | ≥ k for some positive integer k ≥ 3. For every v ∈ V2 , 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 (V2 ) 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 ∈ V2 , 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,
γt ≥ k = |N (V2 )| − |V2 |.
Upon first glance, one may wonder why Graffiti.pc didn’t conjecture instead
that γt (G) ≥ |N (V2 ) − V2 |. Fig. 3.7 shows a 9-vertex counterexample to this
statement, a graph that is definitely in the database given to Graffiti.pc. Note
that |N (V2 ) − V2 | = 5, but γt = 3.
Theorem 3.3.6 is sharp, as can be seen by the following construction. Let k
be a positive integer and let Lk be the graph obtained as follows. Begin with k
copies of K3 and k isolated vertices. From each copy of K3 select two vertices
and add every possible edge between the chosen vertices and the k isolated
vertices (see Figure 3.8 for L2 ). Notice that |N (V2 )| − |V2 | = k and γt (Lk ) = k,
therefore satisfying equality for Theorem 3.3.6.
24
Figure 3.7: 9-vertex counterexample to γt (G) ≥ |N (V2 ) − V2 |.
Figure 3.8: Sharpness of Theorem 3.3.6.
3.4
Total Domination and G2 .
For any graph G, let G2 be the graph with the same vertex set as G, but for
every u, v ∈ V (G) the edge uv ∈ E(G2 ) whenever the distance from u to v in
G is at most two. Clearly any dominating set of G is also a dominating set of
G2 . Hence γ(G) ≥ γ(G2 ). Our next result relates γt (G) to γ(G2 ).
Theorem 3.4.1 (Graffiti.pc). Let G be a graph. Then γt (G) ≥ 2γ(G2 ).
Proof. Let S be a total dominating setP
in G, and let S1 , S2 , . . . , Sk be components of G[S], so that γt (G) = |S| =
|Si |. For each component Si , let Di
be a dominating set of Si and observe that for every v ∈ V (G) there exists a
Sk
vertex u ∈ Di , 1 ≤ i ≤ k such that d(v, u) ≤ 2. It follows that D = i=1 Di is
a dominating set of G2 , and hence γ(G2 ) ≤ |D|. By Theorem 2.1.1,
|D| =
k
X
i=1
k
|Di | ≤
1X
|S|
|Si | =
.
2 i=1
2
25
Therefore, γ(G2 ) ≤ |D| ≤ |S|/2 = γt (G)/2, and rearranging gives γt (G) ≥
2γ(G2 ).
Observe that the bound in Theorem 3.4.1 is sharp for all binary stars. Theorem 3.4.1 has an interesting corollary.
Corollary 3.4.2. Let G be a graph. Then γ(G2 ) ≤ 21 γt (G) ≤ γ(G).
Proof. Recall that the second inequality of Theorem 2.1.3 states that
γt (G) ≤ 2γ(G). By Theorem 3.4.1, we have the following triple inequality:
2γ(G2 ) ≤ γt (G) ≤ 2γ(G).
Dividing the inequality by two yields
γ(G2 ) ≤
1
γt (G) ≤ γ(G).
2
In a moment, we present a companion upper bound for Theorem 3.4.1. We
show that for any graph γt (G) ≤ µ(G) + γ(G2 ). In general, total domination
number and matching number are non-comparable (for example, consider the
graph made up of any number of copies of P4 , each with the same two endpoints).
However, sufficient conditions for upper bounds involving the matching number
have been proven in several papers (see, for example, [8] and [16]). We mention
one such upper bound proven by DeLaViña et al. in [8]. For any graph G, a path
cover is a set of paths such that every vertex of G is an element of exactly one
path. The path covering number, denoted ρ(G), is the cardinality of a smallest
path cover.
Theorem 3.4.3 (DeLaViña et al. [8]). Let G be a graph of order n > 1. Then
γt (G) ≤ µ(G) + ρ(G).
A Hamiltonian path is a path that visits each vertex of a graph exactly once.
It is interesting to note that if a graph G has a Hamiltonian path, then Theorem
3.4.3 says that γt (G) ≤ µ(G)+1. In the following lemma we show that the same
upper bound applies whenever rad(G) ≤ 2. First, we make a basic observation.
Observation 3.4.4. Let G be a graph and let M ⊆ E(G) be a largest matching
of G. Any two unsaturated vertices that are adjacent to an endpoint of the same
matching edge must be adjacent to the same endpoint.
Lemma 3.4.5. Let G be a connected graph with rad(G) ≤ 2. Then γt (G) ≤
µ(G) + 1.
Proof. Let M ⊆ E(G) be a largest matching of G and let v ∈ V (G) be a center
vertex. We identify four types of matching edges relative to v. We call e ∈ M
a 0, 1-edge if v is an endpoint of the edge. If both endpoints of e are adjacent
to v, then e is called a 1, 1-edge. If one endpoint of e is adjacent to v and the
other endpoint is at distance two from v, then we call e a 1, 2-edge. Finally, a
2, 2-edge is a matching edge for which the distance from v to both endpoints of
26
the edge is two. We assume, without loss of generality, that M is chosen in such
a way that v is saturated and M maximizes the number of 1, 2-edges.
We construct a total dominating set S as follows. First, put v in S. Observe
that v dominates both endpoints of a 1, 1-edge. For each 1, 1-edge, we add one
endpoint to S favoring the endpoints adjacent to any unsaturated vertices (each
edge can have only one such endpoint, as noted in Observation 3.4.4; if neither
endpoint is adjacent to any unsaturated vertices, we still add an endpoint to S to
avoid G[S] having any isolated vertices). For every 1, 2-edge, put the endpoint
at distance one from v in S.
Let xy ∈ M be a 2, 2-edge, if any exist. Then there exist intermediary
vertices x0 and y 0 adjacent to v such that x0 ∼ x and y 0 ∼ y. If x0 = y 0 , then
add x0 to S. Suppose x0 6= y 0 . Since M maximizes the number of 1, 2-edges, we
can assume that x0 6∼ y 0 . Thus, by maximality of M and the assumption that
M has a maximum number of 1, 2-edges, there exist e1 , e2 ∈ M for which x0 is
an endpoint of e1 and y 0 is an endpoint of e2 . Since x0 and y 0 are both adjacent
to v, they are endpoints of either a 1, 1-edge or 1, 2-edge.
If either x0 or y 0 is the endpoint of a 1, 2-edge, then it has already been
added to S, as described in the preceding paragraphs. Suppose that x0 or y 0 is
the endpoint of a 1, 1-edge. Since M maximizes number of 1, 2-edges, we can
assume that there are no unsaturated vertices adjacent to an endpoint of the
matching edge for which x0 or y 0 is an endpoint. Thus, if x0 or y 0 are not already
in S, we may swap them with the endpoint that was already in S.
Observe, now, that S dominates every saturated vertex, and that the subgraph induced by S has no isolated vertices. Furthermore, |S| ≤ µ(G) + 1 by
construction. We claim that S is a total dominating set of G. To see this, let u
be an unsaturated vertex. If d(u, v) = 1, u is dominated by v, so suppose that
d(u, v) = 2. Then u is adjacent to a saturated vertex w for which d(w, v) = 1.
By construction, w ∈ S. Thus S is a total dominating set, from which it follows
that γt ≤ |S| ≤ µ(G) + 1.
Let Ŝn be the graph formed by by joining a new vertex to each degree one
vertex of Sn (see Figure 3.9). Then rad(Ŝn ) = 2 and equality for Lemma 3.4.5
is satisfied, but the path covering number may be arbitrarily large.
We will use Lemma 3.4.5 to prove a more general upper bound for all graphs
with no isolated vertices. Recall from Theorem 2.1.2 that every vertex of a
smallest dominating set D has a private neighbor in the complement of D.
Theorem 3.4.6 (Graffiti.pc). Let G be a graph. Then γt (G) ≤ µ(G) + γ(G2 ).
Proof. First, we partition V (G) into γ(G2 ) disjoint subsets T1 , T2 , . . . , Tγ(G2 ) as
follows. Let D be a smallest dominating set of G2 . For each di ∈ D, put di and
all of its private neighbors in Ti . From now on, when we mention the distance
between two vertices we refer to the distance in G. Add to T1 all vertices at
distance two or less from d1 .
For 2 ≤ i ≤ γ(G2 ), add to Ti all vertices at distance two or less from di that
are not already in T1 , T2 , . . . , Ti−1 . If any vertex t ∈ Ti is an isolate of G[Ti ],
then there must exist a vertex t0 ∈ Ti−1 such that t ∼ t0 . Observe that t0 ∼ di
27
Figure 3.9: The graph Ŝ5 .
and d(di−1 , t0 ) = 2, since otherwise t would be an element of Ti−1 . Move t0 from
Ti−1 to Ti , along with any other vertices in V (G) that are adjacent to t0 .
Since each Ti contains di and at least one private neighbor of di , we have
|Ti | ≥ 2 for every i. Clearly,
γ(G2 )
X
µ(G[Ti ]) ≤ µ(G).
i=1
Furthermore, for every i, G[Ti ] is connected and rad(G[Ti ]) ≤ 2. Thus, by
Lemma 3.4.5,
γt (G[Ti ]) ≤ µ(G[Ti ]) + 1.
Moreover,
γ(G2 )
γt (G) ≤
X
γt (G[Ti ])
i=1
from which it follows that,
γ(G2 )
γt (G) ≤
X
i=1
γt (G[Ti ]) ≤
γ(G2 ) X
µ(G[Ti ]) + 1 ≤ µ(G) + γ(G2 ).
i=1
The domination number of G2 is equivalent to what is known as the distance
2-domination number of G (for a complete discussion, see Chapter 12 of [11];
see also [10] and [23]). A set D ⊆ V (G) is a distance 2-dominating set if the
distance between each vertex of V − D and D is at most two. The distance
2-domination number is the cardinality of a smallest distance 2-dominating set.
An interesting corollary to Theorems 3.4.1 and 3.4.6 says, under the preceding
interpretation, that the distance 2-domination number of a graph is at most the
matching number.
28
∆
Figure 3.10: A 16-vertex counterexample to Conjecture 3.5.1.
Corollary 3.4.7. Let G be a graph. Then γ(G2 ) ≤ µ(G).
Proof. By Theorems 3.4.1 and 3.4.6, 2γ(G2 ) ≤ µ(G)+γ(G2 ). Subtracting γ(G2 )
from both sides yields γ(G2 ) ≤ µ(G).
3.5
Miscellaneous conjectures
In this section, several conjectures of Graffiti.pc are presented that did not fit
into any of the preceding sections. All of them are false.
Conjecture 3.5.1 (Graffiti.pc). Let G be a graph. Then γt (G) ≥ µ(G) −
α(G[N (V∆ )]).
False. Fig. 3.10 shows a 16-vertex counterexample to Conjecture 3.5.1. Notice that the graph has a unique vertex of maximum degree, labeled ∆, and
γt (G) = 2. Clearly µ(G) = 8, but α(G[N (V∆ )]) = 5, so µ(G) − α(G[N (V∆ )]) =
3.
Conjecture 3.5.2 (Graffiti.pc). Let G be a graph. Then γt (G) ≥ µ(G[V −
Vδ ]) − γ(G[V − V∆ ]).
False. The graph displayed in Fig. 3.11 is a 21-vertex counterexample to
Conjecture 3.5.2. Notice that the graph has a unique vertex of maximum degree and a unique vertex of minimum degree (labeled ∆ and δ respectively in
Figure 3.11), and that µ(G[V − Vδ ]) = 5 and γ(G[V − V∆ ]) = 2, so the right
hand side of the inequality in Conjecture 3.5.2 is equal to 3, but γt (G) = 2.
Conjecture 3.5.3 (Graffiti.pc). Let G be a graph with ∆ ≤ 3. Then γt (G) ≥
µ(V − Vδ ) − 1.
False. An 18-vertex counterexample is displayed in Fig. 3.12. Notice that
γt = 6, but µ(V − Vδ ) − 1 = 8 − 1 = 7.
29
δ
∆
Figure 3.11: A 21-vertex counterexample to Conjecture 3.5.2.
Conjecture 3.5.4 (Graffiti.pc). Let G be a graph. Then
γt (G) ≥ α(G[V∆ ]) − ∆(G[V∆ ]).
False. Fig. 3.13 presents a 24-vertex cubic counterexample. Every vertex is
a maximum degree vertex. Observe that γt (G) = 8, α(G) = 12 and ∆(G) = 3,
but 8 6≥ 12 − 3 = 9.
For each vertex v of a graph G, the disparity of v is the number of distinct
degrees realized by the neighbors of v. The maximum disparity among all
ˆ
vertices of G is denoted d.
Conjecture 3.5.5 (Graffiti.pc). Let G be a graph with δ ≥ 2. Then γt ≥
ˆ − 1 c), where c is the order of a largest component of the subgraph induced
d(1
2
by vertices with degree at least n2 .
False. A 14-vertex counterexample is shown in Fig. 3.14. A 3-element total
dominating set is marked, so γt ≤ 3. Notice that the maximum degree is 6,
n
and ∆ = 6 < 14
2 = 2 . It follows from Corollary 3.1.1 that γt ≥ 3. Thus
γt = 3 and the set shown is indeed a smallest total dominating set. Observe
that the maximum disparity is 4, and is realized by both vertices u and v labeled
in the figure. Furthermore, there are no vertices with degree at least n2 . But
3 6≥ 4(1 − 0) = 4.
Conjecture 3.5.6 (Graffiti.pc). Let G be a graph. Then γt (G) ≥ ω(G[V2 ]) +
2b 31 γ(G[V3+ ])c, where V3+ is the set of vertices of degree at least three.
False. The graph in Fig 3.14 is also a counterexample to this conjecture.
As noted previously, γt = 3. There are only two vertices of degree two, and
they are adjacent to each other, so ω(G[V2 ]) = 2. The subgraph induced by
the vertices of degree at least 3 is the entire graph, except for the two degree 2
30
Figure 3.12: An 18-vertex counterexample to Conjecture 3.5.3.
Figure 3.13: A 24-vertex counterexample to Conjecture 3.5.4.
vertices. The domination number of this induced subgraph is clearly 3. However,
3 6≥ 2 + 2b 31 (3)c = 4.
Conjecture 3.5.7 (Graffiti.pc). Let G be a graph with δ ≥ 2 and ∆ ≤ n2 .
Then γt ≥ 1 + d1 + 3i e, where i is the number of isolated vertices in the subgraph
induced by the vertices of maximum degree.
False. A 13-vertex counterexample is given in Fig. 3.15. A 3-element total
dominating set is marked in the figure. Since ∆ = 5 < n2 , then γt ≥ 3 by
Corollary 3.1.1. Thus γt = 3. The vertices labeled w, x, y, z are the maximum
degree vertices. Notice that they form an independent set, and are therefore
isolated in the subgraph induced by the maximum degree vertices. Thus i = 4.
But 3 6≥ 1 + d1 + 13 (4)e = 4.
31
u
v
Figure 3.14: A 14-vertex counterexample to Conjectures 3.5.5 and 3.5.6.
w
x
y
z
Figure 3.15: A 13-vertex counterexample to Conjecture 3.5.7.
32
Chapter 4
Concluding Remarks
A total of twenty conjectures of Graffiti.pc were resolved, nine of which are
false. Of the eleven that are true, two are improvements on published results
and are themselves previously unknown. It is also possible that Theorem 3.3.6
and Corollary 3.4.7 are also unknown.
Working with Graffiti.pc provided me with the opportunity to think about
an enormous number of graph invariants, some of which were completely new
to me. This was both fun and challenging. In my opinion, there is no better
way to get to know a subject than to get your hands dirty. Graffiti.pc gave
me plenty of material to play with. For this reason, I believe Graffiti.pc is an
excellent tool for learning and exploring graph theory.
This senior project represents the culmination of nearly eight months of
working with Dr. DeLaViña. It is a gross understatement to say that I have
learned a lot from her. In many ways, she has been my greatest influence at the
University of Houston - Downtown. Since I first met her in my sophomore linear
algebra class, she has played the roles of teacher, mentor, advisor, motivator,
research partner, co-author, counselor and friend. And although this experience
was challenging, time-consuming and often exhausting, I will remember senior
project as one of the highlights of my time spent at UHD.
33
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