Dominating sets and related invariants for graphs
David Amos
Texas A&M University
amosd2@math.tamu.edu
October 16, 2014
David Amos (TAMU)
Dominating sets and related invariants for graphs
October 16, 2014
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What is a graph?
A graph G is a pair of sets (V , E ).
V 6= ∅ is the vertex set; elements are called vertices
E ⊆ {{x, y } : x, y ∈ V } is the edge set; elements called edges
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Dominating sets and related invariants for graphs
October 16, 2014
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What is a graph?
A graph G is a pair of sets (V , E ).
V 6= ∅ is the vertex set; elements are called vertices
E ⊆ {{x, y } : x, y ∈ V } is the edge set; elements called edges
If {x, y } ∈ E :
x and y are adjacent, denoted x ∼ y .
x, y are the endvertices of {x, y }.
David Amos (TAMU)
Dominating sets and related invariants for graphs
October 16, 2014
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What is a graph?
A graph G is a pair of sets (V , E ).
V 6= ∅ is the vertex set; elements are called vertices
E ⊆ {{x, y } : x, y ∈ V } is the edge set; elements called edges
If {x, y } ∈ E :
x and y are adjacent, denoted x ∼ y .
x, y are the endvertices of {x, y }.
|V | is the order of G . If |V | < ∞, then G is finite. Otherwise, G is
infinite. If G is finite, we write n = n(G ) for |V |.
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Dominating sets and related invariants for graphs
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Graph isomorphism
Two graph G and H are isomorphic if there exists a bijection
ϕ : V (G ) → V (H) such that if x ∼ y in G then ϕ(x) ∼ ϕ(y ) in H. We
write G ' H.
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Neighborhoods
For any U ⊆ V , the (open) neighborhood of U is the set
N(U) = y ∈ V : y ∼ x for some x ∈ U .
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Neighborhoods
For any U ⊆ V , the (open) neighborhood of U is the set
N(U) = y ∈ V : y ∼ x for some x ∈ U .
The closed neighborhood of U is the set N[U] = N(U) ∪ U.
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Dominating sets and related invariants for graphs
October 16, 2014
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Neighborhoods
For any U ⊆ V , the (open) neighborhood of U is the set
N(U) = y ∈ V : y ∼ x for some x ∈ U .
The closed neighborhood of U is the set N[U] = N(U) ∪ U.
If U = {x}, we write N(x) and N[x], instead of N({x}) and N[{x}].
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Dominating sets and related invariants for graphs
October 16, 2014
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The degree of a vertex
Let G = (V , E ) be a graph. For each x ∈ V , the degree of x is
deg(x) = |N(x)|.
If deg(x) = deg(y ) for all x, y ∈ V , then G is said to be r -regular, where
r is the common degree of all vertices of G .
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Dominating sets and related invariants for graphs
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Minimum and maximum degree
The minimum degree of a (finite) graph G = (V , E ) is
δ(G ) = min{deg(x) : x ∈ V }.
If deg(x) = ∞ for every x ∈ V , then we write δ(G ) = ∞.
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Dominating sets and related invariants for graphs
October 16, 2014
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Minimum and maximum degree
The minimum degree of a (finite) graph G = (V , E ) is
δ(G ) = min{deg(x) : x ∈ V }.
If deg(x) = ∞ for every x ∈ V , then we write δ(G ) = ∞.
The maximum degree of G is
∆(G ) = max{deg(x) : x ∈ V }.
Note that for a finite graph G with n vertices, ∆(G ) ≤ n − 1.
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Dominating sets and related invariants for graphs
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Paths
A path in a graph G = (V , E ) is a (possibly empty) sequence of edges
e1 = {x1 , y1 }, e2 = {x2 , y2 }, . . . such that xi = yi−1 .
The length of a path is the number of edges contained in the path.
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Dominating sets and related invariants for graphs
October 16, 2014
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Distance
For any x, y ∈ V , the distance between x and y , denoted dist(x, y ) is the
length of a shortest path
e1 = {x, y1 }, e2 = {x2 , y2 }, . . . , et = {xt , y }.
If no such path exists, or if the length of a shortest path is infinite, we set
dist(x, y ) = ∞.
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Dominating sets and related invariants for graphs
October 16, 2014
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Subgraphs
A subgraph of a graph G = (V , E ) as a graph H = (U, F ) such that
U ⊆ V and F ⊆ E .
If U ⊆ V , the subgraph induced by U is the graph G [U] = (U, F ) where
F is just the restriction of E to those edges with both end-vertices in U.
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Dominating sets and related invariants for graphs
October 16, 2014
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Connectedness
A graph is connected if there is a path between any two of its vertices.
That is, if dist(x, y ) < ∞ for any two vertices x, y .
A component of a graph is a maximally connected subgraph.
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Dominating sets and related invariants for graphs
October 16, 2014
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The domination number
Definition
Let G = (V , E ) be a graph. A nonempty subset D ⊆ V is a dominating
set of G if every vertex of G is and element of D or is adjacent to an
element of D. In other words, N[D] = V . The domination number of G ,
denoted γ(G ), is defined as
γ(G ) = min{|D| : N[D] = V }.
If for every dominating set D of G , |D| = ∞, then we set γ(G ) = ∞. It
follows immediately from the definition of γ(G ) that
γ(G ) ≥ # of connected components of G .
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Dominating sets and related invariants for graphs
October 16, 2014
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Example
Let G = (V , E ) be the graph with V = R and
E = {x, y } : x, y ∈ R and |x − y | ∈ Z .
Note that G is disconnected with components consisting of the equivalence
classes of the equivalence relation ∼ defined on R by x ∼ y if and only if
|x − y | ∈ Z. There are infinitely many equivalence classes, so γ(G ) = ∞.
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Dominating sets and related invariants for graphs
October 16, 2014
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Another example
Let G = (V , E ) be the graph defined by V = R and
E = {0, x} : x ∈ R \ {0} .
Then every vertex in V \ {0} is adjacent to 0, so that the set D = {0} is a
dominating set of G . Hence γ(G ) = 1.
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Dominating sets and related invariants for graphs
October 16, 2014
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A finite example: the Peterson graph
The Peterson graph is a 3-regular graph with 10 vertices, which can be
drawn like this:
The domination number of the Peterson graph is 3.
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Dominating sets and related invariants for graphs
October 16, 2014
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Bipartite graphs
A graph G = (V , E ) is bipartite if V can be decomposed into two
disjoint, nonempty subsets A, B ⊂ V (called the parts of V ) so that every
edge of G has one endvertex in A and the other in B.
A complete bipartite graph is a bipartite graph G = (V , E ) with parts A
and B with the property that {x, y } ∈ E if and only if x ∈ A and y ∈ B.
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Dominating sets and related invariants for graphs
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Bipartite graphs
Any two complete bipartite graphs with m + n vertices and parts A and B
with |A| = m and |B| = n are isomorphic.
Hence we refer to the complete bipartite graph with one part of size m and
the other with size n. We denote this graph (actually, isomorphism class)
by Km,n .
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Dominating sets and related invariants for graphs
October 16, 2014
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The domination number of complete bipartite graphs
Here is a drawing of K2,3 .
It is easy to see that γ(K2,3 ) = 2. This is true for all complete bipartite
graphs Km,n with n ≥ m ≥ 2.
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Dominating sets and related invariants for graphs
October 16, 2014
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The domination number of a star
A star on n vertices, denoted Sn , is just the complete bipartite graph
K1,n−1 . By convention, S1 is the graph with one vertex. Here is a drawing
of S5 .
It is easy to see that γ(S5 ) = 1. This is also true for all stars.
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Dominating sets and related invariants for graphs
October 16, 2014
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A characterization of all finite graphs G with γ(G ) = 1
Theorem
Let G be a finite graph. Then γ(G ) = 1 if and only if ∆(G ) = n(G ) − 1.
Proof.
1
By definition, γ(G ) = 1 iff ∃x ∈ V such that N[x] = V .
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Dominating sets and related invariants for graphs
October 16, 2014
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A characterization of all finite graphs G with γ(G ) = 1
Theorem
Let G be a finite graph. Then γ(G ) = 1 if and only if ∆(G ) = n(G ) − 1.
Proof.
1
By definition, γ(G ) = 1 iff ∃x ∈ V such that N[x] = V .
2
N[x] = V if and only if N(x) = V \ {x}.
David Amos (TAMU)
Dominating sets and related invariants for graphs
October 16, 2014
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A characterization of all finite graphs G with γ(G ) = 1
Theorem
Let G be a finite graph. Then γ(G ) = 1 if and only if ∆(G ) = n(G ) − 1.
Proof.
1
By definition, γ(G ) = 1 iff ∃x ∈ V such that N[x] = V .
2
N[x] = V if and only if N(x) = V \ {x}.
3
N(x) = V \ {x} if and only if |N(x)| = n − 1.
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Dominating sets and related invariants for graphs
October 16, 2014
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A characterization of all finite graphs G with γ(G ) = 1
Theorem
Let G be a finite graph. Then γ(G ) = 1 if and only if ∆(G ) = n(G ) − 1.
Proof.
1
By definition, γ(G ) = 1 iff ∃x ∈ V such that N[x] = V .
2
N[x] = V if and only if N(x) = V \ {x}.
3
N(x) = V \ {x} if and only if |N(x)| = n − 1.
4
Since deg(x) = |N(x)|, it follows that deg(x) = n − 1, so that
∆(G ) = n − 1.
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Dominating sets and related invariants for graphs
October 16, 2014
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Covering a graph with stars
A star cover for a graph G = (V , E ) is a family S of vertex-disjoint
subgraphs of G , each member of which is a star, so that every vertex of G
is a vertex of some star in S.
The star cover number of a graph is the cardinality of a smallest star
cover for G .
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Dominating sets and related invariants for graphs
October 16, 2014
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A star cover for the Peterson graph
We can cover the Peterson graph with three stars:
In fact, the star cover number of the Peterson graph is 3.
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Dominating sets and related invariants for graphs
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Central vertices of stars
The central vertex of a star on 3 or more vertices is the unique maximum
degree vertex. For S2 , each vertex is a central vertex; for S1 , there is only
one vertex, which we also call the central vertex.
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Marked star covers
A marked star cover of a graph G is a pair (S, C ), where S is a star
cover of G and C is a set containing exactly one central vertex from each
star in S.
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Marked star covers
A marked star cover of a graph G is a pair (S, C ), where S is a star
cover of G and C is a set containing exactly one central vertex from each
star in S.
Let ∼ be the relation on the set of all marked star covers of a graph G
defined by
(S1 , C1 ) ∼ (S2 , C2 ) if and only if C1 = C2 .
Then ∼ is an equivalence relation. We denote the equivalence class of
(S, C ) by [S, C ], and the set of all equivalence classes of ∼ by M(G ).
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Dominating sets and related invariants for graphs
October 16, 2014
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Dominating sets and marked star covers
Some observations:
If (S, C ) is a marked star cover for a graph G , then C is a dominating
set for G .
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Dominating sets and related invariants for graphs
October 16, 2014
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Dominating sets and marked star covers
Some observations:
If (S, C ) is a marked star cover for a graph G , then C is a dominating
set for G .
If D is a dominating set for G , then the subgraph induced by N[v ] for
each v ∈ D has a subgraph isomorphic to a star.
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Dominating sets and related invariants for graphs
October 16, 2014
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Dominating sets and marked star covers
Some observations:
If (S, C ) is a marked star cover for a graph G , then C is a dominating
set for G .
If D is a dominating set for G , then the subgraph induced by N[v ] for
each v ∈ D has a subgraph isomorphic to a star.
Hence D induces a set of ∼-equivalent marked star covers for G , each
of which has D for a set of central vertices.
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Dominating sets and related invariants for graphs
October 16, 2014
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Dominating sets and marked star covers
Theorem
Let D(G ) be the set of all dominating sets of a graph G . The function
ϕ : D(G ) → M(G ) defined by ϕ(D) = [S, D] is a bijection.
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Dominating sets and related invariants for graphs
October 16, 2014
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Dominating sets and marked star covers
Theorem
Let D(G ) be the set of all dominating sets of a graph G . The function
ϕ : D(G ) → M(G ) defined by ϕ(D) = [S, D] is a bijection.
Corollary
The total number of dominating sets of a graph G is equal to the total
number of distinct sets of central vertices of all star covers for G .
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Dominating sets and related invariants for graphs
October 16, 2014
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Dominating sets and marked star covers
Theorem
Let D(G ) be the set of all dominating sets of a graph G . The function
ϕ : D(G ) → M(G ) defined by ϕ(D) = [S, D] is a bijection.
Corollary
The total number of dominating sets of a graph G is equal to the total
number of distinct sets of central vertices of all star covers for G .
Corollary
For any finite graph, the domination number of G is equal to the star
cover number of G .
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Dominating sets and related invariants for graphs
October 16, 2014
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Some results on the number of dominating sets of a graph
Theorem (A. E. Brouwer, 2009)
The number of dominating sets of a finite graph is odd.
Theorem (D. Bród and Z. Skupień, 2006 (for trees); S. Wagner)
Let G be a connected graph on n > 2 vertices. Then the following
inequality holds:

n/3

n ≡ 0 (mod 3)
5
(n−4)/3
# of dominating sets of G ≥ 9 · 5
n ≡ 1 (mod 3)


(n−2)/3
3·5
n ≡ 2 (mod 3)
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Dominating sets and related invariants for graphs
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Why study dominating sets?
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Dominating sets and related invariants for graphs
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Why study dominating sets?
1. Many practical problems involving scheduling tasks and assigning jobs
and resources are problems about dominating sets. There are many
applications in computer science and computer engineering.
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Dominating sets and related invariants for graphs
October 16, 2014
27 / 1
Why study dominating sets?
1. Many practical problems involving scheduling tasks and assigning jobs
and resources are problems about dominating sets. There are many
applications in computer science and computer engineering.
2. The dominating set decision problem is a classical NP-complete
problem: Given a graph G and input K , determine if γ(G ) ≤ K . So if
P 6= NP, there is no efficient algorithm for finding a smallest dominating
set for a given graph.
David Amos (TAMU)
Dominating sets and related invariants for graphs
October 16, 2014
27 / 1
Why study dominating sets?
1. Many practical problems involving scheduling tasks and assigning jobs
and resources are problems about dominating sets. There are many
applications in computer science and computer engineering.
2. The dominating set decision problem is a classical NP-complete
problem: Given a graph G and input K , determine if γ(G ) ≤ K . So if
P 6= NP, there is no efficient algorithm for finding a smallest dominating
set for a given graph.
An enormous body of literature exists on domination in graphs, including
two not-so-small books by T. Haynes, S. Hedetniemi and P. Slater.
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Dominating sets and related invariants for graphs
October 16, 2014
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An upper bound for the domination number
Theorem (Ore’s Theorem, 1960)
For any finite graph G with no isolated vertices, γ(G ) ≤ n2 .
For the proof we need, the following definition: A subset I of vertices of a
graph G is an independent set if no two vertices of I are adjacent.
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Dominating sets and related invariants for graphs
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An upper bound for the domination number
Theorem (Ore’s Theorem, 1960)
For any finite graph G with no isolated vertices, γ(G ) ≤ n2 .
For the proof we need, the following definition: A subset I of vertices of a
graph G is an independent set if no two vertices of I are adjacent.
PROOF SKETCH: A largest independent set I is a dominating set, and
since G has no isolated vertices, so is V (G ) − I . So
γ(G ) ≤ min{I , V − I } ≤ n2 .
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Another upper bound for the domination number
Definition
The size of a largest independent set of a graph G is called the
independence number of G , and is denoted α(G ).
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Dominating sets and related invariants for graphs
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Another upper bound for the domination number
Definition
The size of a largest independent set of a graph G is called the
independence number of G , and is denoted α(G ).
Corollary (To the proof of Ore’s Theorem)
For any graph G , γ(G ) ≤ α(G ).
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Dominating sets and related invariants for graphs
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Another upper bound for the domination number
Definition
The size of a largest independent set of a graph G is called the
independence number of G , and is denoted α(G ).
Corollary (To the proof of Ore’s Theorem)
For any graph G , γ(G ) ≤ α(G ).
The independence number decision problem is another classical
NP-complete problem, so this bound is not computable and hence not as
desirable as the previous bound.
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Dominating sets and related invariants for graphs
October 16, 2014
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An improved upper bound for the domination number
Theorem (Folklore)
For any finite graph G , γ(G ) ≤
n
∆(G )+1 .
PROOF: Each vertex in a dominating set has at most ∆(G ) neighbors
that are not in the dominating set. Hence n − γ(G ) ≤ γ(g )∆(G ), which
after rearranging gives γ(G ) ≤ ∆(Gn)+1 .
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Dominating sets and related invariants for graphs
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An improved upper bound for the domination number
Theorem (Folklore)
For any finite graph G , γ(G ) ≤
n
∆(G )+1 .
PROOF: Each vertex in a dominating set has at most ∆(G ) neighbors
that are not in the dominating set. Hence n − γ(G ) ≤ γ(g )∆(G ), which
after rearranging gives γ(G ) ≤ ∆(Gn)+1 .
This bound is sharp for stars and complete graphs (in fact, any graph with
∆(G ) = n − 1). Note that when ∆(G ) ≥ 1 we have γ(G ) ≤ ∆(Gn)+1 ≤ n2 .
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Dominating sets and related invariants for graphs
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Domatic partitions
Let G = (V , E ). A domatic partition of G is a partition of V into
disjoint sets V1 , V2 , . . . , Vk such that Vi is a dominating set for G for all i.
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The domatic number
Definition
Let G = (V , E ). The domatic number of G is the maximum size of a
domatic partition of G .
For example, the domatic number of the Peterson graph is at least 2:
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An upper bound for the domatic number
Theorem (Folklore)
The domatic number of a graph G is at most δ(G ) + 1.
PROOF: Consider N[v ] for some minimum degree vertex v . Every
dominating set of a domatic partition contains a vertex in N[v ]. Moreover,
each vertex in N[v ] is contained in at most one dominating set in the
partition. Hence any domatic partition has size at most |N[v ]| = δ(G ) + 1.
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A lower bound for the domatic number
Theorem (Folklore)
For any graph G with no isolated vertices, the domatic number of G is at
least 2. In other words, any graph with no isolated vertices can be
partitioned into 2 disjoint dominating sets.
PROOF: Follows from the proof of Ore’s Theorem.
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A “fundamental” theorem
Let G = (V , E ) be a graph. For any subset S ⊆ V and v ∈ S, we say that
a vertex u ∈ V − S adjacent to v is a private neighbor of v (relative to
S) if u has no other neighbors in S.
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A “fundamental” theorem
Let G = (V , E ) be a graph. For any subset S ⊆ V and v ∈ S, we say that
a vertex u ∈ V − S adjacent to v is a private neighbor of v (relative to
S) if u has no other neighbors in S.
Theorem (B. Bollobás and E. Cockayne, 1979)
For any graph G = (V , E ) with no isolated vertices, there exists a smallest
dominating set D for G such that every vertex of D has a private neighbor
in V − D.
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A corollary of the “fundamental” theorem
A matching in a graph G is a set of edges, no two of which have a
common endvertex. The matching number of G , denoted µ(G ), is the
size of a largest matching.
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A corollary of the “fundamental” theorem
A matching in a graph G is a set of edges, no two of which have a
common endvertex. The matching number of G , denoted µ(G ), is the
size of a largest matching.
Corollary
For any graph G with no isolated vertices, γ(G ) ≤ µ(G ).
PROOF SKETCH: The set of edges between each dominating vertex and
one of their private neighbors is a matching in G .
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Dominating sets and related invariants for graphs
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A corollary of the “fundamental” theorem
A matching in a graph G is a set of edges, no two of which have a
common endvertex. The matching number of G , denoted µ(G ), is the
size of a largest matching.
Corollary
For any graph G with no isolated vertices, γ(G ) ≤ µ(G ).
PROOF SKETCH: The set of edges between each dominating vertex and
one of their private neighbors is a matching in G .
Note that µ(G ) ≤ n2 , so this is another improvement of Ore’s Theorem.
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Total dominating sets
Definition
A total dominating set for a graph G is a dominating set D for G with
the property that every vertex in D has a neighbor in D. Note that total
dominating sets are not defined for graphs with isolated vertices.
The total domination number of G , denoted γt (G ), is the cardinality of
a smallest total dominating set for G .
By definition, γt (G ) ≥ γ(G ) and γt (G ) ≥ 2.
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Example: Peterson Graph Again
A 4-element total dominating set for the Peterson graph:
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A lower bound for the total domination number
Theorem (Folklore)
For any finite graph G = (V , E ), γt (G ) ≥
n(G )
∆(G ) .
PROOF: Each vertex x in a smallest dominating set D can have at most
deg(x) − 1 neighbors in V \ D. So
X
n − |D| ≤
(deg(x) − 1) ≤ (∆(G ) − 1)|D|.
x∈D
Rearranging gives γt (G ) = |D| ≥
David Amos (TAMU)
n(G )
∆(G ) .
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Upper bounds for the total domination number
Theorem (Folklore)
For any graph G with no isolated vertices, γt (G ) ≤ 2γ(G ).
PROOF: Follows from the fundamental theorem.
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Upper bounds for the total domination number
Theorem (Folklore)
For any graph G with no isolated vertices, γt (G ) ≤ 2γ(G ).
PROOF: Follows from the fundamental theorem.
Theorem (E. Cockayne, R. Dawes, S. Hedetniemi, 1980)
For any finite graph G = (V , E ), γt (G ) ≤ 32 n(G ).
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Complexity and other results
The total domination decision problems was shown to be NP-complete in
1983 by by J. Pfaff, R. Laskar and S. Hedetniemi.
Although NP-complete in general, the problem is linear in trees (R. Laskar,
J. Pfaff, S. M. Hedetniemi and S. T. Hedetniemi, 1984).
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Complexity and other results
The total domination decision problems was shown to be NP-complete in
1983 by by J. Pfaff, R. Laskar and S. Hedetniemi.
Although NP-complete in general, the problem is linear in trees (R. Laskar,
J. Pfaff, S. M. Hedetniemi and S. T. Hedetniemi, 1984).
In 2009, M. Henning published a 30-page survey of results on the total
domination number. A monograph on total domination in graphs was
published by Hennings and A. Yeo in 2013.
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Other flavors of domination
Let G = (V , E ) be a graph. Here are some other types of dominating sets.
Connected dominating set: A dominating set D for which which
induces a connected subgraph; the connected domination number
is denoted γc (G ).
David Amos (TAMU)
Dominating sets and related invariants for graphs
October 16, 2014
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Other flavors of domination
Let G = (V , E ) be a graph. Here are some other types of dominating sets.
Connected dominating set: A dominating set D for which which
induces a connected subgraph; the connected domination number
is denoted γc (G ).
Independent dominating set: A dominating set D that is also an
independent set; the independent domination number is denoted
γi (G ).
David Amos (TAMU)
Dominating sets and related invariants for graphs
October 16, 2014
42 / 1
Other flavors of domination
Let G = (V , E ) be a graph. Here are some other types of dominating sets.
Connected dominating set: A dominating set D for which which
induces a connected subgraph; the connected domination number
is denoted γc (G ).
Independent dominating set: A dominating set D that is also an
independent set; the independent domination number is denoted
γi (G ).
k-Distance dominating set: A subset D ⊆ V such that for every
y ∈ D − V there exists an x ∈ D such that dist(x, y ) ≤ k. Note that
a 1-distance dominating set is the same as a dominating set.
David Amos (TAMU)
Dominating sets and related invariants for graphs
October 16, 2014
42 / 1
Other flavors of domination
Let G = (V , E ) be a graph. Here are some other types of dominating sets.
Connected dominating set: A dominating set D for which which
induces a connected subgraph; the connected domination number
is denoted γc (G ).
Independent dominating set: A dominating set D that is also an
independent set; the independent domination number is denoted
γi (G ).
k-Distance dominating set: A subset D ⊆ V such that for every
y ∈ D − V there exists an x ∈ D such that dist(x, y ) ≤ k. Note that
a 1-distance dominating set is the same as a dominating set.
k-Dominating set: A dominating set D such that ever vertex in D
has at least k neighbors in V \ D. The k-domination number is
denoted γk (G ). Note that γ1 (G ) = γ(G ).
David Amos (TAMU)
Dominating sets and related invariants for graphs
October 16, 2014
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The Cartesian product of two graphs
Let G and H be any two graphs. The Cartesian product of G and H,
denoted G H, is the graph with vertex set V (G ) × V (H) and edge set
defined by the following relation:
(u, v ) ∼ (u 0 , v 0 ), where u, u 0 ∈ V (G ) and v , v 0 ∈ V (H) if and only if
either:
u 0 = v 0 and u ∼ v in G , or
u = v and u 0 ∼ v 0 in H.
David Amos (TAMU)
Dominating sets and related invariants for graphs
October 16, 2014
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Example
From Wikipedia:
David Amos (TAMU)
Dominating sets and related invariants for graphs
October 16, 2014
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Some facts about Cartesian products
The operation (G , H) 7→ G H is associative:
(G H) K = G (H K ).
In general, the operation is not commutative.
David Amos (TAMU)
Dominating sets and related invariants for graphs
October 16, 2014
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Some facts about Cartesian products
The operation (G , H) 7→ G H is associative:
(G H) K = G (H K ).
In general, the operation is not commutative.
The set of all (isomorphism classes of) finite graphs is a monoid under the
Cartesian product; the identity is the graph with a single vertex.
David Amos (TAMU)
Dominating sets and related invariants for graphs
October 16, 2014
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Vizing’s Conjecture
Conjecture (Vizing, 1968)
For any two graph G and G , γ(G H) ≥ γ(G )γ(H).
The conjecture holds when either G and H are both stars.
David Amos (TAMU)
Dominating sets and related invariants for graphs
October 16, 2014
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Vizing’s Conjecture
Conjecture (Vizing, 1968)
For any two graph G and G , γ(G H) ≥ γ(G )γ(H).
The conjecture holds when either G and H are both stars.
It also holds if only one of G or H is a star.
David Amos (TAMU)
Dominating sets and related invariants for graphs
October 16, 2014
46 / 1
Vizing’s Conjecture
Conjecture (Vizing, 1968)
For any two graph G and G , γ(G H) ≥ γ(G )γ(H).
The conjecture holds when either G and H are both stars.
It also holds if only one of G or H is a star.
In 2000, W. Clark and S. Suen shows that for any two graphs G and H,
γ(G H) ≥ 12 γ(G )γ(H).
David Amos (TAMU)
Dominating sets and related invariants for graphs
October 16, 2014
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References
B. Bollobás and E. J. Cockayne, Graph-theoretic parameters concerning domination,
independence, and irredundance, J. Graph Theory 3 (1979), 214–249.
D. Bród and Z. Skupień, Trees with extremal numbers of dominating sets. Australas. J.
Combin. 35 (2006), 273-290.
A. E. Brouwers, P. Csorba and A. Schrijver, The number of dominating sets of a finite
graph is odd. Preprint, 2009.
W. E. Clark and S. Suen, An inequality related to Vizings conjecture, Electron. J.
Combin. 7 (2000).
E. J. Cockayne, R. M. Dawes and S. T. Hedetniemi, Total domination in graphs,
Networks 10 (1980), 211–219.
T. W. Haynes, S. T. Hedetniemi, and P. Slater, Fundamentals of domination in graphs.
Monographs and Textbooks in Pure and Applied Mathematics, 208. Marcel Dekker,
Inc., New York, 1998.
David Amos (TAMU)
Dominating sets and related invariants for graphs
October 16, 2014
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References
T. W. Haynes, S. T. Hedetniemi, and P. Slater, Domination in graphs: advanced topics
Monographs and Textbooks in Pure and Applied Mathematics, 208. Marcel Dekker,
Inc., New York, 1998.
M. Henning, A survey of selected recent results on total domination in graphs, Discrete
Mathematics 309 (2009), 32–63.
R. C. Laskar, J. Pfaff, S. M. Hedetniemi and S. T. Hedetniemi, On the algorithmic
complexity of total domination, SIAM J. Algebraic Discrete Methods 5 (1984),420–425.
Ø. Ore, Note on Hamilton circuits, American Mathematical Monthly 67 (1960), 55.
J. Pfaff, R. C. Laskar and S. T. Hedetniemi, NP-completeness of total and connected
domination and irredundance for bipartite graphs, Technical Report 428, Clemson
University, Dept. of Math. Sciences, 1983.
V. G. Vizing, Some unsolved problems in graph theory, Uspehi Mat. Naukno. (in
Russian) 23 (1968), 117–134.
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Dominating sets and related invariants for graphs
October 16, 2014
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