1/20/2015
Ma/CS 6b
Class 7: Minors
By Adam Sheffer
Edge Subdivision

Given a graph 𝐺 = 𝑉, 𝐸 , and an edge
𝑒 ∈ 𝐸, subdividing 𝑒 is the operation of
replacing 𝑒 with a path consisting of new
vertices.
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Graph Relations

We saw two types of relations between
graphs 𝐺 and 𝐻:
◦ 𝐻 is a subgraph of 𝐺.
◦ 𝐻 is an induced subgraph of 𝐺.

We say that 𝐺 is a subdivision of 𝐻 if 𝐺 is
obtained by subdividing some (or all) of
the edges of 𝐻.
Graph Subdivision
𝐻
𝐺
𝐺 is a subdivision of 𝐻.
 We refer to the added vertices as
subdivision vertices.

◦ These vertices are of degree 2.
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Topological Minors

A graph 𝐻 is a topological minor of a
graph 𝐺 if 𝐺 contains a subdivision of 𝐻
as a subgraph.
𝐺
𝐻
Vertex Suppression

Given a graph 𝐺 = 𝑉, 𝐸 , and a vertex
𝑣 ∈ 𝑉 of degree 2, suppressing 𝑣 is the
operation of removing 𝑣 and adding an
edge between the two neighbors of 𝑉.
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An Equivalent Definition

A graph 𝐻 is a topological minor of a
graph 𝐺 if 𝐻 can be obtained from 𝐺 by
suppressing vertices (of degree 2) and by
removing edges and vertices.
𝐻
𝐺
No Order

Notice that the order of the vertex and
edge removal and of the vertex
suppression does not matter (we will not formally
prove this).
𝐻
𝐺
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Test Your Intuition

Is 𝐻 a topological minor of 𝐺?
𝐺

𝐻
No, since 𝐻 contains a vertex of degree 4
but 𝐺 does not.
Test Your Intuition #2

Is 𝐾5 a topological minor of the Petersen
graph?
The Petersen
graph

No. As before, we cannot create vertices
of degree 4 by deletions and
suppressions.
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Edge Contraction

Given a graph 𝐺 = 𝑉, 𝐸 , and an edge
𝑒 ∈ 𝐸, contracting 𝑒 is the operation of
removing 𝑒 and merging its two
endpoints.
◦ We merge any resulting parallel edges.
𝑎
𝑓
𝑐
𝑏
𝑎
𝑑
𝑒
𝑏
𝑓
𝑒
𝑐/𝑑
Minors

A graph 𝐻 is a minor of a graph 𝐺 if 𝐻 can
be obtained from 𝐺 by contracting edges
and by removing edges and vertices.
𝑎
𝑏
𝑎
𝑐
𝐺
𝑏
𝑐
𝐻
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Test Your Intuition #3

Is 𝐻 a minor of 𝐺?
𝐺

𝐻
Yes!
Test Your Intuition #4

Is 𝐾5 a minor of the Petersen graph?
The Petersen
graph

Yes. Contract the five edges that connect
an inner vertex to an outer vertex.
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An Alternative Point of View

Let 𝐻 be a minor of 𝐺. Then every vertex
of 𝐻 corresponds to a connected
subgraph of 𝐺.
𝐺
𝐻
No Order #2

When performing deletions and
contractions, the order of the deletions
does not matter (we will not formally prove this).
𝐺
𝐻
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Etymology

What is the meaning of the word matrix?
◦ Wikipedia: “coined by James Joseph Sylvester
in 1850 who understood a matrix as an object
giving rise to a number of determinants today
called minors”. Sylvester explains:
◦ “I have in previous papers defined a "Matrix"
as a rectangular array of terms, out of which
different systems of determinants may be
engendered as from the womb of a
common parent.”
No 𝐶3 Minors

Problem. Characterize the family of
graphs that do not have 𝐶3 as a minor.

Solution. These are exactly the graphs
that do not contain any cycles.
◦ If a graph contains a cycle 𝐶𝑘 , we can remove
everything except for this 𝐶𝑘 and then
contract it to 𝐶3 .
◦ If a graph contains no cycles, we cannot
obtain a cycle by contracting and removing
edges and vertices.
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No 𝐶4 Minors

Problem. Characterize the family of
graphs that do not have 𝐶4 as a minor.

Solution. These are exactly the graphs
that do not contain cycles of length ≥ 4.
◦ If a graph contains a cycle 𝐶𝑘 with 𝑘 ≥ 4, we
can remove everything except for this 𝐶𝑘 and
then contract it to 𝐶4 .
◦ If a graph does not contain 𝐶𝑘 with 𝑘 ≥ 4, we
cannot obtain a 𝐶4 by contracting and
removing edges and vertices.
Minors and Topological Minors

Prove or disprove. Every topological
minor of a graph 𝐺 is also a minor of 𝐺.
◦ A minor is obtained by removing edges and
vertices and contracting edges.
◦ A topological minor is obtained by removing
edges and vertices and by suppressing
vertices.
◦ The claim is true since vertex suppression can
be seen as a special case of edge contraction.
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Connections Between Graph
Relations

Warm up. Which relation is stronger?
◦ 𝐻 is a subgraph of 𝐺.
◦ 𝐻 is an induced subgraph of 𝐺.

An induced subgraph is also a subgraph,
but the opposite is not always true.
𝐺
𝐻
Connections Between Graph
Relations

What are the connections between the
following relations?
◦
◦
◦
◦
◦
𝐻 is a subgraph of 𝐺.
𝐻 is an induced subgraph of 𝐺.
𝐻 is minor of 𝐺.
𝐻 is a topological minor of 𝐺.
𝐺 is a subdivision of 𝐻.
Subdiv.
Topo.
⊂ Minor ⊂
Minor
Induced
⊂ Subgraph
Subgraph
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Connections (cont.)

There is no relation between
◦ 𝐻 is a subgraph of 𝐺.
◦ 𝐺 is a subdivision of 𝐻.
Subdivision and
not a subgraph
𝐺
Subgraph and
No subdivision
Connections (cont.)
Deletions and
suppressions
Only
suppressions
Subdiv.
⊂ Topo.
Induced
⊂ Subgraph
Subgraph
Minor
Vertex
deletions
Only
deletions
⊂
⊂
Minor
Deletions and
contractions
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Minors and Bounded Degrees

Claim. Consider 𝐺 = 𝑉, 𝐸 , and let 𝐻 be
a minor of 𝐺 with maximum degree
three. Then 𝐻 is also a topological minor
of 𝐺.
Proof
𝐻 is obtained from 𝐺 by removing edges
and vertices and contracting edges.
 We need to show that 𝐻 can be obtained
from 𝐺 by removing edges and vertices
and by suppressing vertices.

◦ We first perform all of the deletions.
◦ We then need to handle contracting edges.
◦ After performing the deletions, we may
assume that no vertex has degree more than
three, also after performing any number of
the contractions.
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Proof: Two Cases

We consider contracting an edge 𝑢, 𝑣 .
◦ If either 𝑢 or 𝑣 is of degree 1, this is just a
vertex removal.
𝑣
𝑢
𝑣
◦ If either 𝑢 or 𝑣 is of degree 2, this can be
replaced with vertex suppression.
𝑤
𝑢
𝑣
𝑤
𝑣
Proof: Another Case

We consider the contraction of an edge 𝑒
between vertices 𝑢, 𝑣.
◦ Consider the case where both 𝑢 and 𝑣 are of
degree 3 and with a common neighbor 𝑤.
𝑤
𝑤
𝑢
𝑣
◦ We delete the edge 𝑣, 𝑤 and then suppress
𝑣.
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Proof: The Last Case

We consider the contraction of an edge 𝑒
between vertices 𝑢, 𝑣.
◦ Consider the case where both 𝑢 and 𝑣 are of
degree 3 with no common neighbors.
𝑢
𝑣
◦ This contraction yields a vertex of degree 4.
Contradiction! (since we performed the
deletions first to avoid such a case)
The End
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