Basic notation
Vingzing Conjecture
Results
Domination Number of Cartesian Product
of Graphs
Advisor : Xuding Zhu
Wen Wang
National Kaohsiung Normal University, 2013,8,10
Examples
Basic notation
Vingzing Conjecture
Outline
1
Basic notation
2
Vingzing Conjecture
Hisotry
BG-graphs
Double projection
3
Results
Theorem1
Theorem2
4
Examples
Range of k ?
Critical graphs
Results
Examples
Basic notation
Vingzing Conjecture
Results
Examples
Let G = (V , E) be a simple graph with vertex set V (G) and
edge set E(G). For any vertex v ∈ V (G), the neighborhood of
v is the vertex set N(v ) = {u | u is adjacent to v } and the close
neighborhood of v is the vertex set N[v ] = N(v ) ∪ {v }.
For example, N(v1 ) = {v2 , v5 } and N[v1 ] = {v1 , v2 , v5 }.
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Vingzing Conjecture
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Def.
For D ⊆ V
[(G), D is a dominating set of G if
N[v ].
V (G) =
v ∈D
The domination number of G is the minimum cardinality of
a dominating set of G , we denote it by γ(G).
For example, S
V (C5 ) ⊆ N[v1 ] N[v3 ] and it’s
impossible to find a vertex to
dominate C5 , γ(C5 ) = 2.
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Basic notation
Vingzing Conjecture
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Examples
Def.
γ(G) = min {|D| : V (G) ⊆
D⊆V (G)
[
N[v ]}.
v ∈D
For S is a subgraph of G,
[
γG (S) = min {|D| : V (S) ⊆
N[v ]}.
D⊆V (G)
v ∈D
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Let S = G − {v4 } − {v
S8 } − {v9 }.
Since V (S) ⊆ N[v6 ] N[v5 ],
γG (S) = 2.
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Vingzing Conjecture
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Def.
For two graphs G and H, the Cartesian Product G2H is
the graph with vertex set {(u, v )|u ∈ v (G), v ∈ V (H)} and
(u, v )(u 0 , v 0 ) ∈ E(G2H) whenever u = u 0 and vv 0 ∈ E(H)
or v = v 0 and uu 0 ∈ E(G).
Figure on the below side show a example of P4 2P4 and
P4 2P4 2K2 .
Examples
Basic notation
Vingzing Conjecture
Results
In 1963, V.G.Vizing conjectured that for any graph G and H,
γ(G2H) ≥ γ(G)γ(H).
Examples
Basic notation
Vingzing Conjecture
Results
Examples
Hisotry
Theorem
(RON, TIBOR, ) If a simple graph G is path, tree, cycle, chordal
graph or γ(G) ≤ 3 then for any simple graph H
γ(G2H) ≥ γ(G)γ(H).
Basic notation
Vingzing Conjecture
Results
Examples
BG-graphs
Theorem
(Barcalkin and German )
If V (G) can be covered by γ(G) complete subgraphs , then for
every graph H,
γ(G2H) ≥ γ(G)γ(H).
Basic notation
Vingzing Conjecture
Results
Double projection
Theorem
(Clark and Suen )
γ(G2H) ≥
for any graphs G and H.
1
γ(G)γ(H)
2
Examples
Basic notation
Vingzing Conjecture
Results
Examples
Theorem1
Theorem
If a simple graph G has k disjoint complete subgraphs
k
S
S1 , S2 , ...Sk such that γG ( Si ) = k , then for any simple graph
i=1
H
γ(G2H) ≥ k γ(H).
Basic notation
Vingzing Conjecture
Results
Examples
Theorem2
Theorem
Let S1 , S2 , .., Sk be disjoint sets of V (G). If for any D ⊆ V (G),
we have |D| + |BD | ≥ k where BD = { i |Si 6⊆ N[D], Si ∩ D = ∅},
then
γ(H2G) ≥ k γ(H) for any simple graph H.
𝑆1
𝑆2
D
𝑆3
𝑆4
𝑆5
𝑆6
V(G) – N[D]
𝐵𝐷 = { 5 , 6 }
Basic notation
Vingzing Conjecture
Results
Examples
Range of k ?
Def.
A set X ⊆ V (G) is called a 2-packing if d(u, v ) > 2 for any
different vertices u and v of X . The 2-pakcing number ρ2 (G) is
the maximum order of a 2-packing of G.
Proposition
γ(G) ≥ k ≥ ρ2 (G) for any graph G.
Basic notation
Vingzing Conjecture
Results
Range of k ?
Def.
Let I be an independent set of vertices in the simple graph G,
the least size of a set of vertices in G that dominates I is
denoted by γI (G), and we denote the largest γI (G) over all
independent sets I in G by γ i (G).
Proposition
k ≥ γ i (G) for any graph G.
PS. γ(G) = γ i (G) for any chordal graph G.
Examples
Basic notation
Vingzing Conjecture
Results
Range of k ?
Is it possible that k = γ(G) for every graph ?
Examples
Basic notation
Vingzing Conjecture
Results
Examples
Critical graphs
Def.
G is called γ1 -k -critical graph if
k = γ(G) > γ(G + e) for any e ∈ E(Gc ).
G is called γ2 -k -critical graph if
k = γ(G) < γ(G − e) for any e ∈ E(G).
Theorem
If G is a γ1 -k-critical graph and it has S1 , S2 , . . . , Sk disjoint
sets of V (G) such that for any D ⊆ V (G), |D| + |BD | ≥ k , then
Si is complete for all i.
Basic notation
Vingzing Conjecture
Results
Critical graphs
Figure: γ1 -3-critical graphs
Examples
Basic notation
Vingzing Conjecture
Results
Critical graphs
Theorem
Let D = {v1 , v2 , . . . vk } be a dominating set of a γ2 -k-critical
graph G, then for any k disjoint complete subgraphs
k
S
G1 , G2 , . . . Gk with vi ∈ V (Gi ) for all i, γG ( Gi ) = k .
i=1
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Basic notation
Vingzing Conjecture
Results
Critical graphs
Every γ1 -k -critical graph can be constructed by adding edges
to some γ2 -k-critical graphs.
𝐾𝑛
………….
ϒ -(k-1)-critical graph
2
?
ϒ -k-critical graph
1
+ edges
ϒ -k-critical graph
2
Is it possible we can add edges to any γ1 -k-critical graph to
construct a γ2 -(k-1)-critical graph ?
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Basic notation
Vingzing Conjecture
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Critical graphs
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γ1 -V (G)-critical
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Vingzing Conjecture
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Critical graphs
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γ1 -7-critical
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Vingzing Conjecture
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Critical graphs
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γ1 -6-critical
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Vingzing Conjecture
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Critical graphs
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γ1 -5-critical
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Vingzing Conjecture
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Critical graphs
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γ1 -4-critical
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Vingzing Conjecture
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Critical graphs
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γ1 -3-critical
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Vingzing Conjecture
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Critical graphs
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γ1 -2-critical
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Vingzing Conjecture
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Critical graphs
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Kn
Examples
Basic notation
Critical graphs
Thank you.
Vingzing Conjecture
Results
Examples