Introduction to Line Graphs

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Introduction to Line Graphs

• Emphasizing their construction, clique
  decompositions, and regularity

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??
A given graph X
Its line graph L(X)
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This vertex has degree 1.
so it gives no new adjacencies.
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This vertex has degree 1.
so it gives no new adjacencies.
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This vertex also has degree 1.
so it gives no new adjacencies.
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This vertex has degree 0.
so it gives no new adjacencies.
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The given graph X
Its line graph L(X)
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The given graph X
Its line graph L(X)
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Definition of Line Graph

• Def. Given a graph X(V,E), its line graph is the
  graph L(X) with vertex set E and where two
  distinct vertices
• e and e
• are adjacent iff the corresponding edges in graph
  X share an endpoint.

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Vertices of degree 2
Cliques partitioning edge set
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Note Each vertex is in 2 of these
cliques.
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• Thm. (Krausz 1943) Let X be a nonempty simple
  graph. Then
• X is the line graph of some graph
• if and only if
• the edges of X can be partitioned into cliques so
  that each vertex belongs to 2.

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2
2
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3
2
2
2
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A vertex in only 1 clique is a pendant. (A deg
0 vertex is an isolated edge.)
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P is ?-free, so maximal cliques have size 2.
So to cover the edges, each vertex must be in 3
cliques.
So P cannot be a line graph.
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P is ?-free, so maximal cliques have size 2.
Same problem. So P cannot be a line graph.
So to cover the edges, each vertex must be in 3
cliques.
So P cannot be a line graph.
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Isolated vertices must correspond to isolated
edges.
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Each clique in the partition corresponds to a
vertex.
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Two vertices are adjacent iff their cliques
shared a vertex.
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Vertices belonging to only a single clique
correspond to pendants
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L(X)
X
Success!!
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Determining whether such a clique decomposition
exists is computationally impractical for large
graphs.
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Determining whether such a clique decomposition
exists is computationally impractical for large
graphs. Beineke and Lehot developed a better
approach in the 60s 70s using results of van
Rooij and Wilf.
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Determining whether such a clique decomposition
exists is computationally impractical for large
graphs. Beineke and Lehot developed a better
approach in the 60s 70s using results of van
Rooij and Wilf.
Theorem. (Beineke, 1968) A simple graph is a
line graph if and only if it has none of the
following as an induced subgraph
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Alternatively, we could state the result as
follows
Theorem. (Beineke, 1968) A simple graph is a
line graph if and only each induced subgraph on ?
6 vertices is a line graph.
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Alternatively, we could state the result as
follows
Theorem. (Beineke, 1968) A simple graph is a
line graph if and only each induced subgraph on ?
6 vertices is a line graph.
This narrows the search and in fact, there now
exists a linear time algorithm to test whether a
graph is a line graph! (It also produces such a
graph when the answer is affirmative.)
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Non-uniqueness of the inverse problem.
This graph is the line graph of each of
the following
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Non-uniqueness of the inverse problem.
This graph is the line graph of each of
the following
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Non-uniqueness of the inverse problem.
This graph is the line graph of each of
the following
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Non-uniqueness of the inverse problem.
This graph is the line graph of each of
the following
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Non-uniqueness of the inverse problem.
This graph is the line graph of each of
the following
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BUT the problem is not any worse than these
examples illustrate.
K3
K1,3
Star-Triangle Issue (or claw-triangle)
Theorem. (Whitney, 1932) K3 and K1.3 are the
ONLY pair of non-isomorphic connected graphs
whose line graphs are isomorphic.
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Finally, some results about regularity and line
graphs.
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Finally, some results about regularity and line
graphs.

• If X is regular with valency k, then L(X) is
  regular with valency 2k-2.

k-1
k-1
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Finally, some results about regularity and line
graphs.

• If X is regular with valency k, then L(X) is
  regular with valency 2k-2.

k-1
k-1

• If X is connected and L(X) is regular, then
  either
• (i) X is regular, or
• (ii) X is bipartite semi-regular.

a
b
x
y
k
a b k 2 same for any yx.
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Review of topics

• Definition of line graphs
• Cliques partitioning edges of L(X) using vertices
  in X of degree 2
• Beinekes 9 forbidden subgraphs
• Non-uniqueness of inverse problem
• Whitneys star-triangle result
• Regularity and Semiregularity in X and L(X)

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The End