graph2 - basics.sjtu.edu.cnbasics.sjtu.edu.cn/~chen/teaching/gr08/graph2.pdf · graph theory (ii)...
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GRAPH THEORY
Yijia ChenShanghai Jiaotong University
2008/2009
Shanghai
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GRAPH THEORY (II) Page 2
1.4 Connectivity
Shanghai
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GRAPH THEORY (II) Page 2
1.4 Connectivity
A non-empty graph G is called connected if any two of its vertices are linked by a path in G.
Shanghai
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GRAPH THEORY (II) Page 2
1.4 Connectivity
A non-empty graph G is called connected if any two of its vertices are linked by a path in G.
If U ⊆ V (G) and G[U ] is connected, we also call U itself connected (in G).
Shanghai
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GRAPH THEORY (II) Page 2
1.4 Connectivity
A non-empty graph G is called connected if any two of its vertices are linked by a path in G.
If U ⊆ V (G) and G[U ] is connected, we also call U itself connected (in G).
Instead of ‘not connected’ we usually say ‘disconnected’.
Shanghai
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GRAPH THEORY (II) Page 2
1.4 Connectivity
A non-empty graph G is called connected if any two of its vertices are linked by a path in G.
If U ⊆ V (G) and G[U ] is connected, we also call U itself connected (in G).
Instead of ‘not connected’ we usually say ‘disconnected’.
Proposition. The vertices of a connected graph G can always be enumerated, say as
v1, . . . , vn, so that Gi := G[v1, . . . , vi] is connected for every i.
Shanghai
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GRAPH THEORY (II) Page 3
Let G = (V, E) be a graph. A maximal connected subgraph of G is called a component of
G.
Shanghai
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GRAPH THEORY (II) Page 3
Let G = (V, E) be a graph. A maximal connected subgraph of G is called a component of
G.
note: A component, being connected, is always non-empty: the empty graph, therefore, has
no components.
Shanghai
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GRAPH THEORY (II) Page 4
Given sets A, B of vertices, We call P = x0 . . . xk an A-B path if
V (P ) ∩ A = {x0} and V (P ) ∩ B = {xk}.
Shanghai
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GRAPH THEORY (II) Page 4
Given sets A, B of vertices, We call P = x0 . . . xk an A-B path if
V (P ) ∩ A = {x0} and V (P ) ∩ B = {xk}.
If A, B ⊆ V and X ⊆ V ∪E are such that every A-B path in G contains a vertex or an edge
from X , we say that X separates A and B in G.
Shanghai
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GRAPH THEORY (II) Page 4
Given sets A, B of vertices, We call P = x0 . . . xk an A-B path if
V (P ) ∩ A = {x0} and V (P ) ∩ B = {xk}.
If A, B ⊆ V and X ⊆ V ∪E are such that every A-B path in G contains a vertex or an edge
from X , we say that X separates A and B in G.
note: This implies A ∩ B ⊆ X
Shanghai
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GRAPH THEORY (II) Page 4
Given sets A, B of vertices, We call P = x0 . . . xk an A-B path if
V (P ) ∩ A = {x0} and V (P ) ∩ B = {xk}.
If A, B ⊆ V and X ⊆ V ∪E are such that every A-B path in G contains a vertex or an edge
from X , we say that X separates A and B in G.
note: This implies A ∩ B ⊆ X
The unordered pair {A, B} is a separation of G if A ∪ B = V (G) and G has no edge
between A \ B and B \ A.
Shanghai
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GRAPH THEORY (II) Page 4
Given sets A, B of vertices, We call P = x0 . . . xk an A-B path if
V (P ) ∩ A = {x0} and V (P ) ∩ B = {xk}.
If A, B ⊆ V and X ⊆ V ∪E are such that every A-B path in G contains a vertex or an edge
from X , we say that X separates A and B in G.
note: This implies A ∩ B ⊆ X
The unordered pair {A, B} is a separation of G if A ∪ B = V (G) and G has no edge
between A \ B and B \ A.
Clearly, the latter is equivalent to saying that A ∩ B separates A from B.
Shanghai
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GRAPH THEORY (II) Page 4
Given sets A, B of vertices, We call P = x0 . . . xk an A-B path if
V (P ) ∩ A = {x0} and V (P ) ∩ B = {xk}.
If A, B ⊆ V and X ⊆ V ∪E are such that every A-B path in G contains a vertex or an edge
from X , we say that X separates A and B in G.
note: This implies A ∩ B ⊆ X
The unordered pair {A, B} is a separation of G if A ∪ B = V (G) and G has no edge
between A \ B and B \ A.
Clearly, the latter is equivalent to saying that A ∩ B separates A from B.
If both A \ B and B \ A are non-empty, the separation is proper.
Shanghai
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GRAPH THEORY (II) Page 4
Given sets A, B of vertices, We call P = x0 . . . xk an A-B path if
V (P ) ∩ A = {x0} and V (P ) ∩ B = {xk}.
If A, B ⊆ V and X ⊆ V ∪E are such that every A-B path in G contains a vertex or an edge
from X , we say that X separates A and B in G.
note: This implies A ∩ B ⊆ X
The unordered pair {A, B} is a separation of G if A ∪ B = V (G) and G has no edge
between A \ B and B \ A.
Clearly, the latter is equivalent to saying that A ∩ B separates A from B.
If both A \ B and B \ A are non-empty, the separation is proper.
The number |A ∩ B| is the order of the separation {A, B}.
Shanghai
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GRAPH THEORY (II) Page 5
If U is any set of vertices (usually of G), we write G − U for G[V \ U ].
Shanghai
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GRAPH THEORY (II) Page 5
If U is any set of vertices (usually of G), we write G − U for G[V \ U ].
In other words, G − U is obtained from G by deleting all the vertices in U ∩ V and their
incident edges.
Shanghai
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GRAPH THEORY (II) Page 5
If U is any set of vertices (usually of G), we write G − U for G[V \ U ].
In other words, G − U is obtained from G by deleting all the vertices in U ∩ V and their
incident edges.
We say that X separates G if G − X is disconnected, that is, if X separates in G some two
vertices that are not in X .
Shanghai
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GRAPH THEORY (II) Page 6
G is called k-connected (for k ∈ N) if |G| > k and G − X is connected for every set
X ⊆ V with |X | < k.
Shanghai
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GRAPH THEORY (II) Page 6
G is called k-connected (for k ∈ N) if |G| > k and G − X is connected for every set
X ⊆ V with |X | < k.
In other words, no two vertices of G are separated by fewer than k other vertices.
Shanghai
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GRAPH THEORY (II) Page 6
G is called k-connected (for k ∈ N) if |G| > k and G − X is connected for every set
X ⊆ V with |X | < k.
In other words, no two vertices of G are separated by fewer than k other vertices.
Every (non-empty) graph is 0-connected, and the 1-connected graphs are precisely the
non-trivial connected graphs.
Shanghai
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GRAPH THEORY (II) Page 6
G is called k-connected (for k ∈ N) if |G| > k and G − X is connected for every set
X ⊆ V with |X | < k.
In other words, no two vertices of G are separated by fewer than k other vertices.
Every (non-empty) graph is 0-connected, and the 1-connected graphs are precisely the
non-trivial connected graphs.
The greatest integer k such that G is k-connected is the connectivity κ(G) of G.
Shanghai
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GRAPH THEORY (II) Page 6
G is called k-connected (for k ∈ N) if |G| > k and G − X is connected for every set
X ⊆ V with |X | < k.
In other words, no two vertices of G are separated by fewer than k other vertices.
Every (non-empty) graph is 0-connected, and the 1-connected graphs are precisely the
non-trivial connected graphs.
The greatest integer k such that G is k-connected is the connectivity κ(G) of G.
For nonempty G, κ(G) = 0 if and only G is disconnected or a K1.
Shanghai
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GRAPH THEORY (II) Page 6
G is called k-connected (for k ∈ N) if |G| > k and G − X is connected for every set
X ⊆ V with |X | < k.
In other words, no two vertices of G are separated by fewer than k other vertices.
Every (non-empty) graph is 0-connected, and the 1-connected graphs are precisely the
non-trivial connected graphs.
The greatest integer k such that G is k-connected is the connectivity κ(G) of G.
For nonempty G, κ(G) = 0 if and only G is disconnected or a K1.
κ(Kn) = n − 1 for all n ≥ 1.
Shanghai
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GRAPH THEORY (II) Page 7
For a subset F of [V ]2 we write G − F := (V, E \ F ) and G + F := (V, E ∪ F ).
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GRAPH THEORY (II) Page 7
For a subset F of [V ]2 we write G − F := (V, E \ F ) and G + F := (V, E ∪ F ).
If |G| > 1 and G − F is connected for every set F ⊆ E of fewer than � edges, then G is
called �-edge-connected.
Shanghai
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GRAPH THEORY (II) Page 7
For a subset F of [V ]2 we write G − F := (V, E \ F ) and G + F := (V, E ∪ F ).
If |G| > 1 and G − F is connected for every set F ⊆ E of fewer than � edges, then G is
called �-edge-connected.
The greatest integer � such that G is �-edge-connected is the edge-connectivity λ(G) of G.
If particular, we have λ(G) = 0 if G is disconnected.
Shanghai
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GRAPH THEORY (II) Page 7
For a subset F of [V ]2 we write G − F := (V, E \ F ) and G + F := (V, E ∪ F ).
If |G| > 1 and G − F is connected for every set F ⊆ E of fewer than � edges, then G is
called �-edge-connected.
The greatest integer � such that G is �-edge-connected is the edge-connectivity λ(G) of G.
If particular, we have λ(G) = 0 if G is disconnected.
Proposition. If G is non-trivial then κ(G) ≤ λ(G) ≤ δ(G).
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GRAPH THEORY (II) Page 8
High connectivity requires a large minimum degree. Conversely, large minimum degree
does not ensure high connectivity, not even high edge-connectivity.
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GRAPH THEORY (II) Page 8
High connectivity requires a large minimum degree. Conversely, large minimum degree
does not ensure high connectivity, not even high edge-connectivity.
Theorem.[Mader 1972] Let 0 �= k ∈ N. Every graph G with d(G) ≥ 4k has a(k + 1)-connected subgraph H such that ε(H) > ε(G) − k.
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GRAPH THEORY (II) Page 9
Proof.
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GRAPH THEORY (II) Page 9
Proof.
Let γ := ε(G)(≥ 2k).
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GRAPH THEORY (II) Page 9
Proof.
Let γ := ε(G)(≥ 2k).
Consider the subgraphs G′ ⊆ G such that
|G′| ≥ 2k and ‖G′‖ > γ(|G′| − k). (1)
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GRAPH THEORY (II) Page 9
Proof.
Let γ := ε(G)(≥ 2k).
Consider the subgraphs G′ ⊆ G such that
|G′| ≥ 2k and ‖G′‖ > γ(|G′| − k). (1)
Let H be one of the smallest order.
Shanghai
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GRAPH THEORY (II) Page 9
Proof.
Let γ := ε(G)(≥ 2k).
Consider the subgraphs G′ ⊆ G such that
|G′| ≥ 2k and ‖G′‖ > γ(|G′| − k). (1)
Let H be one of the smallest order. No G′ as in (1) can have order exactly 2k; otherwise
‖G′‖ > γk ≥ 2k2 >
(|G′|2
).
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GRAPH THEORY (II) Page 9
Proof.
Let γ := ε(G)(≥ 2k).
Consider the subgraphs G′ ⊆ G such that
|G′| ≥ 2k and ‖G′‖ > γ(|G′| − k). (1)
Let H be one of the smallest order. No G′ as in (1) can have order exactly 2k; otherwise
‖G′‖ > γk ≥ 2k2 >
(|G′|2
).
The minimality of H then implies δ(H) > γ; otherwise we could delete a vertex of degree
at most γ and obtain a graph G” of smaller order satisfying (1).
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GRAPH THEORY (II) Page 9
Proof.
Let γ := ε(G)(≥ 2k).
Consider the subgraphs G′ ⊆ G such that
|G′| ≥ 2k and ‖G′‖ > γ(|G′| − k). (1)
Let H be one of the smallest order. No G′ as in (1) can have order exactly 2k; otherwise
‖G′‖ > γk ≥ 2k2 >
(|G′|2
).
The minimality of H then implies δ(H) > γ; otherwise we could delete a vertex of degree
at most γ and obtain a graph G” of smaller order satisfying (1).
Hence |H| ≥ γ, thus ‖H‖ > γ(|H| − k) implies
ε(H) > γ − k.
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GRAPH THEORY (II) Page 10
Proof. [Cont’d]
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GRAPH THEORY (II) Page 10
Proof. [Cont’d]
It remains to show that H is (k + 1)-connected.
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GRAPH THEORY (II) Page 10
Proof. [Cont’d]
It remains to show that H is (k + 1)-connected.
If not, then H has a proper separation {U1, U2} of order at most k.
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GRAPH THEORY (II) Page 10
Proof. [Cont’d]
It remains to show that H is (k + 1)-connected.
If not, then H has a proper separation {U1, U2} of order at most k. Let Hi := H[Ui] for
i = 1, 2.
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GRAPH THEORY (II) Page 10
Proof. [Cont’d]
It remains to show that H is (k + 1)-connected.
If not, then H has a proper separation {U1, U2} of order at most k. Let Hi := H[Ui] for
i = 1, 2. Since any vertex v ∈ U1 \ U2 has all its d(v) ≥ δ(H) > γ neighbours from H in
H1, we have |H1| ≥ γ ≥ 2k.
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GRAPH THEORY (II) Page 10
Proof. [Cont’d]
It remains to show that H is (k + 1)-connected.
If not, then H has a proper separation {U1, U2} of order at most k. Let Hi := H[Ui] for
i = 1, 2. Since any vertex v ∈ U1 \ U2 has all its d(v) ≥ δ(H) > γ neighbours from H in
H1, we have |H1| ≥ γ ≥ 2k. Similarly, |H2| ≥ 2k.
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GRAPH THEORY (II) Page 10
Proof. [Cont’d]
It remains to show that H is (k + 1)-connected.
If not, then H has a proper separation {U1, U2} of order at most k. Let Hi := H[Ui] for
i = 1, 2. Since any vertex v ∈ U1 \ U2 has all its d(v) ≥ δ(H) > γ neighbours from H in
H1, we have |H1| ≥ γ ≥ 2k. Similarly, |H2| ≥ 2k.
By the minimality of H with respect to (1), we have
‖Hi‖ ≤ γ(|Hi| − k)
for i = 1, 2.
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GRAPH THEORY (II) Page 10
Proof. [Cont’d]
It remains to show that H is (k + 1)-connected.
If not, then H has a proper separation {U1, U2} of order at most k. Let Hi := H[Ui] for
i = 1, 2. Since any vertex v ∈ U1 \ U2 has all its d(v) ≥ δ(H) > γ neighbours from H in
H1, we have |H1| ≥ γ ≥ 2k. Similarly, |H2| ≥ 2k.
By the minimality of H with respect to (1), we have
‖Hi‖ ≤ γ(|Hi| − k)
for i = 1, 2. But then
‖H‖ ≤ ‖H1‖ + ‖H2‖≤ γ(|H1| + |H2| − 2k)
≤ γ(|H| − k) (as |H1 ∩ H2| ≤ k),
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GRAPH THEORY (II) Page 10
Proof. [Cont’d]
It remains to show that H is (k + 1)-connected.
If not, then H has a proper separation {U1, U2} of order at most k. Let Hi := H[Ui] for
i = 1, 2. Since any vertex v ∈ U1 \ U2 has all its d(v) ≥ δ(H) > γ neighbours from H in
H1, we have |H1| ≥ γ ≥ 2k. Similarly, |H2| ≥ 2k.
By the minimality of H with respect to (1), we have
‖Hi‖ ≤ γ(|Hi| − k)
for i = 1, 2. But then
‖H‖ ≤ ‖H1‖ + ‖H2‖≤ γ(|H1| + |H2| − 2k)
≤ γ(|H| − k) (as |H1 ∩ H2| ≤ k),
which contradicts (1) for H . �
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GRAPH THEORY (II) Page 11
1.5 Trees and forests
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GRAPH THEORY (II) Page 11
1.5 Trees and forests
An acyclic graph, one not containing any cycles, is called a forest.
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GRAPH THEORY (II) Page 11
1.5 Trees and forests
An acyclic graph, one not containing any cycles, is called a forest.
A connected forest is called a tree.
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GRAPH THEORY (II) Page 11
1.5 Trees and forests
An acyclic graph, one not containing any cycles, is called a forest.
A connected forest is called a tree.
The vertices of degree 1 in a tree are its leaves,
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GRAPH THEORY (II) Page 11
1.5 Trees and forests
An acyclic graph, one not containing any cycles, is called a forest.
A connected forest is called a tree.
The vertices of degree 1 in a tree are its leaves, except that the root of a tree is never called a
leaf, even if it has degree 1.
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GRAPH THEORY (II) Page 12
Theorem. The following assertions are equivalent for a graph T :
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GRAPH THEORY (II) Page 12
Theorem. The following assertions are equivalent for a graph T :
(i) T is a tree;
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GRAPH THEORY (II) Page 12
Theorem. The following assertions are equivalent for a graph T :
(i) T is a tree;
(ii) Any two vertices of T are linked by a unique path in T ;
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GRAPH THEORY (II) Page 12
Theorem. The following assertions are equivalent for a graph T :
(i) T is a tree;
(ii) Any two vertices of T are linked by a unique path in T ;
(iii) T is minimally connected, i.e. T is connected but T − e is disconnected for every edge
e ∈ T ;
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GRAPH THEORY (II) Page 12
Theorem. The following assertions are equivalent for a graph T :
(i) T is a tree;
(ii) Any two vertices of T are linked by a unique path in T ;
(iii) T is minimally connected, i.e. T is connected but T − e is disconnected for every edge
e ∈ T ;
(iv) T is maximally acyclic, i.e. T contains no cycle but T + xy does, for any two
non-adjacent vertices x, y ∈ T .
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GRAPH THEORY (II) Page 12
Theorem. The following assertions are equivalent for a graph T :
(i) T is a tree;
(ii) Any two vertices of T are linked by a unique path in T ;
(iii) T is minimally connected, i.e. T is connected but T − e is disconnected for every edge
e ∈ T ;
(iv) T is maximally acyclic, i.e. T contains no cycle but T + xy does, for any two
non-adjacent vertices x, y ∈ T .
We write xTy for the unique path in a tree T between two vertices x and y.
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GRAPH THEORY (II) Page 12
Theorem. The following assertions are equivalent for a graph T :
(i) T is a tree;
(ii) Any two vertices of T are linked by a unique path in T ;
(iii) T is minimally connected, i.e. T is connected but T − e is disconnected for every edge
e ∈ T ;
(iv) T is maximally acyclic, i.e. T contains no cycle but T + xy does, for any two
non-adjacent vertices x, y ∈ T .
We write xTy for the unique path in a tree T between two vertices x and y.
Every connected graph contains a spanning tree: by the equivalence of (i) and (iii), any
minimal connected spanning subgraph will be a tree.
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GRAPH THEORY (II) Page 13
Corollary. The vertices of a tree can always be enumerated, say as v1, . . . , vn, so that every
vi with i ≥ 2 has a unique neighbour in {v1, . . . , vi−1}.
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GRAPH THEORY (II) Page 13
Corollary. The vertices of a tree can always be enumerated, say as v1, . . . , vn, so that every
vi with i ≥ 2 has a unique neighbour in {v1, . . . , vi−1}.
Corollary. A connected graph with n vertices is a tree if and only if it has n − 1 edges.
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GRAPH THEORY (II) Page 13
Corollary. The vertices of a tree can always be enumerated, say as v1, . . . , vn, so that every
vi with i ≥ 2 has a unique neighbour in {v1, . . . , vi−1}.
Corollary. A connected graph with n vertices is a tree if and only if it has n − 1 edges.
Corollary. If T is a tree and G is any graph with δ(G) ≥ |T | − 1, then T ⊆ G, i.e., G has a
subgraph isomorphic to T .
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GRAPH THEORY (II) Page 14
Sometimes it is convenient to consider one vertex of a tree as special;
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GRAPH THEORY (II) Page 14
Sometimes it is convenient to consider one vertex of a tree as special; such a vertex is then
called the root of this tree.
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GRAPH THEORY (II) Page 14
Sometimes it is convenient to consider one vertex of a tree as special; such a vertex is then
called the root of this tree.
A tree T with a fixed root r is a rooted tree.
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GRAPH THEORY (II) Page 14
Sometimes it is convenient to consider one vertex of a tree as special; such a vertex is then
called the root of this tree.
A tree T with a fixed root r is a rooted tree.
Writing x ≤ y for x ∈ rTy then defines a partial ordering on V (T ), the tree-order
associated with T and r.
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GRAPH THEORY (II) Page 14
Sometimes it is convenient to consider one vertex of a tree as special; such a vertex is then
called the root of this tree.
A tree T with a fixed root r is a rooted tree.
Writing x ≤ y for x ∈ rTy then defines a partial ordering on V (T ), the tree-order
associated with T and r.
We shall think of this ordering as expressing height: if x < y we say that x lies below y in
T .
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GRAPH THEORY (II) Page 14
Sometimes it is convenient to consider one vertex of a tree as special; such a vertex is then
called the root of this tree.
A tree T with a fixed root r is a rooted tree.
Writing x ≤ y for x ∈ rTy then defines a partial ordering on V (T ), the tree-order
associated with T and r.
We shall think of this ordering as expressing height: if x < y we say that x lies below y in
T .
We call
y� :={x | x ≤ y
}and �x :=
{y | y ≥ x
}
the down-closure of y and the up-closure of x.
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GRAPH THEORY (II) Page 15
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GRAPH THEORY (II) Page 15
- The root r is the least element in this partial order.
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GRAPH THEORY (II) Page 15
- The root r is the least element in this partial order.
- The leaves of T are its maximal elements.
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GRAPH THEORY (II) Page 15
- The root r is the least element in this partial order.
- The leaves of T are its maximal elements.
- The ends of any edge of T are comparable.
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GRAPH THEORY (II) Page 15
- The root r is the least element in this partial order.
- The leaves of T are its maximal elements.
- The ends of any edge of T are comparable.
- The down-closure of every vertex is a chain, a set of pairwise comparable elements.
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GRAPH THEORY (II) Page 15
- The root r is the least element in this partial order.
- The leaves of T are its maximal elements.
- The ends of any edge of T are comparable.
- The down-closure of every vertex is a chain, a set of pairwise comparable elements.
- The vertices at distance k from r have height k and form the kth level of T .
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GRAPH THEORY (II) Page 16
Given a graph H , we call a path P an H-path if P is non-trivial and meets H exactly in its
ends.
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GRAPH THEORY (II) Page 16
Given a graph H , we call a path P an H-path if P is non-trivial and meets H exactly in its
ends.
In particular, the edge of any H-path of length 1 is never an edge of H .
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GRAPH THEORY (II) Page 16
Given a graph H , we call a path P an H-path if P is non-trivial and meets H exactly in its
ends.
In particular, the edge of any H-path of length 1 is never an edge of H .
A rooted tree T contained in a graph G is called normal in G if the ends of every T -path in
G are comparable in the tree-order of T .
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GRAPH THEORY (II) Page 16
Given a graph H , we call a path P an H-path if P is non-trivial and meets H exactly in its
ends.
In particular, the edge of any H-path of length 1 is never an edge of H .
A rooted tree T contained in a graph G is called normal in G if the ends of every T -path in
G are comparable in the tree-order of T .
If T spans G, this amounts to requiring that two vertices of T must be comparable whenever
they are adjacent in G.
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GRAPH THEORY (II) Page 17
Lemma. Let T be a normal tree in G.
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GRAPH THEORY (II) Page 17
Lemma. Let T be a normal tree in G.
(i) Any two vertices x, y ∈ T are separated in G by the set x� ∩ y�.
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GRAPH THEORY (II) Page 17
Lemma. Let T be a normal tree in G.
(i) Any two vertices x, y ∈ T are separated in G by the set x� ∩ y�.
(ii) If S ⊆ V (T ) = V (G) and S is down-closed, then the components of G − S are spanned
by the sets �x with x minimal in T − S.
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GRAPH THEORY (II) Page 18
Normal spanning trees are also called depth-first search trees.
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GRAPH THEORY (II) Page 18
Normal spanning trees are also called depth-first search trees.
Proposition. Every connected graph contains a normal spanning tree, with any specified
vertex as its root.
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GRAPH THEORY (II) Page 19
1.6 Bipartite graphs
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GRAPH THEORY (II) Page 19
1.6 Bipartite graphs
Let r ≥ 2 be an integer. A graph G = (V, E) is called r-partite if V admits a partition into r
classes such that every edge has its ends in different classes:
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GRAPH THEORY (II) Page 19
1.6 Bipartite graphs
Let r ≥ 2 be an integer. A graph G = (V, E) is called r-partite if V admits a partition into r
classes such that every edge has its ends in different classes: vertices in the same partition
class must not be adjacent.
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GRAPH THEORY (II) Page 19
1.6 Bipartite graphs
Let r ≥ 2 be an integer. A graph G = (V, E) is called r-partite if V admits a partition into r
classes such that every edge has its ends in different classes: vertices in the same partition
class must not be adjacent.
Instead of ‘2-partite’ one usually says bipartite.
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GRAPH THEORY (II) Page 19
1.6 Bipartite graphs
Let r ≥ 2 be an integer. A graph G = (V, E) is called r-partite if V admits a partition into r
classes such that every edge has its ends in different classes: vertices in the same partition
class must not be adjacent.
Instead of ‘2-partite’ one usually says bipartite.
Theorem. A graph is bipartite if and only if it contains no odd cycle.
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GRAPH THEORY (II) Page 20
1.7 Contraction and minors
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GRAPH THEORY (II) Page 20
1.7 Contraction and minors
Let e = xy be an edge of a graph G = (V, E). By G/e we denote the graph obtained from
G by contracting the edge e into a new vertex ve, which becomes adjacent to all the former
neighbours of x and of y.
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GRAPH THEORY (II) Page 20
1.7 Contraction and minors
Let e = xy be an edge of a graph G = (V, E). By G/e we denote the graph obtained from
G by contracting the edge e into a new vertex ve, which becomes adjacent to all the former
neighbours of x and of y.
Formally, G/e is a graph (V ′, E′) with vertex set V ′ := (V \ {x, y})∪̇{ve} (where ve is the
‘new’ vertex, i.e., ve /∈ V ∪ E) and edge set
E′ :={
vw ∈ E∣∣ {v, w} ∩ {x, y} = ∅
}
∪{
vew∣∣ xw ∈ E \ {e} or yw ∈ E \ {e}
}.
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GRAPH THEORY (II) Page 21
More generally, if X is another graph and {Vx|x ∈ V (X)} is a partition of V into connected
subsets such that, for any two vertices x, y ∈ X , there is a Vx-Vy edge in G if and only if
xy ∈ E(X), we call G an MX and write G = MX .
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GRAPH THEORY (II) Page 21
More generally, if X is another graph and {Vx|x ∈ V (X)} is a partition of V into connected
subsets such that, for any two vertices x, y ∈ X , there is a Vx-Vy edge in G if and only if
xy ∈ E(X), we call G an MX and write G = MX .
The sets Vx are the branch sets of this MX .
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GRAPH THEORY (II) Page 21
More generally, if X is another graph and {Vx|x ∈ V (X)} is a partition of V into connected
subsets such that, for any two vertices x, y ∈ X , there is a Vx-Vy edge in G if and only if
xy ∈ E(X), we call G an MX and write G = MX .
The sets Vx are the branch sets of this MX .
Intuitively, we obtain X from G by contracting every branch set to a single vertex and
deleting any ‘parallel edges’ or ‘loops’ that may arise.
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GRAPH THEORY (II) Page 22
Proposition. G is an MX if and only if X can be obtained from G by a series of edge
contractions, i.e. if and only if there are graphs G0, . . . , Gn and edges ei ∈ Gi such that
G0 = G, Gn = X , and Gi+1 = Gi/ei for all i < n.
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GRAPH THEORY (II) Page 23
If G = MX is a subgraph of another graph Y , we call X a minor of Y and write X � Y .
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GRAPH THEORY (II) Page 23
If G = MX is a subgraph of another graph Y , we call X a minor of Y and write X � Y .
- Every subgraph of a graph is also its minor; in particular, every graph is its own minor.
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GRAPH THEORY (II) Page 23
If G = MX is a subgraph of another graph Y , we call X a minor of Y and write X � Y .
- Every subgraph of a graph is also its minor; in particular, every graph is its own minor.
- Any minor of a graph can be obtained from it by first deleting some vertices and edges, and
then contracting some further edges.
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GRAPH THEORY (II) Page 23
If G = MX is a subgraph of another graph Y , we call X a minor of Y and write X � Y .
- Every subgraph of a graph is also its minor; in particular, every graph is its own minor.
- Any minor of a graph can be obtained from it by first deleting some vertices and edges, and
then contracting some further edges.
- Conversely, any graph obtained from another by repeated deletions and contractions (in
any order) is its minor.
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GRAPH THEORY (II) Page 24
If we replace the edges of X with independent paths between their ends (so that none of
these paths has an inner vertex on another path or in X), we call the graph G obtained a
subdivision of X and write G = TX .
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GRAPH THEORY (II) Page 24
If we replace the edges of X with independent paths between their ends (so that none of
these paths has an inner vertex on another path or in X), we call the graph G obtained a
subdivision of X and write G = TX .
If G = TX is a subgraph of another graph Y , then X is a topological minor of Y .
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GRAPH THEORY (II) Page 24
If we replace the edges of X with independent paths between their ends (so that none of
these paths has an inner vertex on another path or in X), we call the graph G obtained a
subdivision of X and write G = TX .
If G = TX is a subgraph of another graph Y , then X is a topological minor of Y .
If G = TX , we view V (X) as a subset of V (G) and call these vertices the branch vertices
of G; the other vertices of G are its subdividing vertices.
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GRAPH THEORY (II) Page 24
If we replace the edges of X with independent paths between their ends (so that none of
these paths has an inner vertex on another path or in X), we call the graph G obtained a
subdivision of X and write G = TX .
If G = TX is a subgraph of another graph Y , then X is a topological minor of Y .
If G = TX , we view V (X) as a subset of V (G) and call these vertices the branch vertices
of G; the other vertices of G are its subdividing vertices.
All subdividing vertices have degree 2, while the branch vertices retain their degree from X .
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GRAPH THEORY (II) Page 25
Proposition. (i) Every TX is also an MX ; thus, every topological minor of a graph is also
its (ordinary) minor.
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GRAPH THEORY (II) Page 25
Proposition. (i) Every TX is also an MX ; thus, every topological minor of a graph is also
its (ordinary) minor.
(ii) If Δ(X) ≤ 3, then every MX contains a TX ; thus, every minor with maximum degree
at most 3 of a graph is also its topological minor.
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GRAPH THEORY (II) Page 25
Proposition. (i) Every TX is also an MX ; thus, every topological minor of a graph is also
its (ordinary) minor.
(ii) If Δ(X) ≤ 3, then every MX contains a TX ; thus, every minor with maximum degree
at most 3 of a graph is also its topological minor.
Proposition. The minor relation � and the topological-minor relation are partial orderingson the class of finite graphs, i.e. they are reflexive, antisymmetric and transitive.
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GRAPH THEORY (II) Page 26
1.8 Euler tours
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GRAPH THEORY (II) Page 26
1.8 Euler tours
A walk (of length k) in a graph G is a non-empty alternating sequence
v0e0v1e1 . . . ek−1vk
of vertices and edges in G such that ei = {vi, vi+1} for all i < k.
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GRAPH THEORY (II) Page 26
1.8 Euler tours
A walk (of length k) in a graph G is a non-empty alternating sequence
v0e0v1e1 . . . ek−1vk
of vertices and edges in G such that ei = {vi, vi+1} for all i < k.
If v0 = vk, the walk is closed.
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GRAPH THEORY (II) Page 26
1.8 Euler tours
A walk (of length k) in a graph G is a non-empty alternating sequence
v0e0v1e1 . . . ek−1vk
of vertices and edges in G such that ei = {vi, vi+1} for all i < k.
If v0 = vk, the walk is closed.
If the vertices in a walk are all distinct, it defines an path in G.
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GRAPH THEORY (II) Page 26
1.8 Euler tours
A walk (of length k) in a graph G is a non-empty alternating sequence
v0e0v1e1 . . . ek−1vk
of vertices and edges in G such that ei = {vi, vi+1} for all i < k.
If v0 = vk, the walk is closed.
If the vertices in a walk are all distinct, it defines an path in G.
In general, every walk between two vertices contains a path between these vertices.
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GRAPH THEORY (II) Page 27
A closed walk in a graph is an Euler tour if it traverses every edge of the graph exactly once.
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GRAPH THEORY (II) Page 27
A closed walk in a graph is an Euler tour if it traverses every edge of the graph exactly once.
A graph is Eulerian if it admits an Euler tour.
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GRAPH THEORY (II) Page 27
A closed walk in a graph is an Euler tour if it traverses every edge of the graph exactly once.
A graph is Eulerian if it admits an Euler tour.
Theorem.[Euler 1736] A connected graph is Eulerian if and only if every vertex has even
degree.
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