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Introduction to Line Graphs Emphasizing their construction, clique decompositions, and regularity

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A given graph XIts 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 XIts line graph L(X)

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The given graph XIts 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 ≥ 2Cliques 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. For an exposition of the proof, see West.

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3 2 2 2 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 Δ-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 Δ-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. Same problem. 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 share 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 60’s & 70’s 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 60’s & 70’s 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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This graph is the line graph of each of the following: Non-uniqueness of the inverse problem….

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This graph is the line graph of each of the following: Non-uniqueness of the inverse problem….

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This graph is the line graph of each of the following: Non-uniqueness of the inverse problem….

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This graph is the line graph of each of the following: Non-uniqueness of the inverse problem….

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This graph is the line graph of each of the following: Non-uniqueness of the inverse problem….

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K3K3 BUT… the problem is not any worse than these examples illustrate. K 1,3 Theorem. (Whitney, 1932) K 3 and K 1.3 are the ONLY pair of non- isomorphic connected graphs whose line graphs are isomorphic. “Star-Triangle” Issue (or “claw-triangle”)

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Finally, some results about regularity and line graphs….

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1.If X is regular (valency k), then L(X) is regular (valency 2k-2). L(X) k-1 regular X

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Finally, some results about regularity and line graphs…. 1.If X is regular (valency k), then L(X) is regular (valency 2k-2). 2.If L(X) is regular, then the components of X are regular or bipartite semi-regular. k-1 k regular L(X) regular X

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Finally, some results about regularity and line graphs…. k-1 deg(x) deg(y) k deg(x) + deg(y) = k + 2 same for any y~x. xy regular L(X) 1.If X is regular (valency k), then L(X) is regular (valency 2k-2). 2.If L(X) is regular, then the components of X are regular or bipartite semi-regular. regular X

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All must have same degree as y. xy k regular L(X)

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deg(z) = deg(y) xy z k regular L(X)

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deg(a) = deg(x) xy z a k regular L(X)

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degrees must alternate! xy z a d b f e c g k regular L(X) If X is connected: there can be at most 2 degrees in X. if X has an odd cycle, X is regular. else X is bipartite and semi-regular. … Therefore …

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Two results about regularity and line graphs 1.If X is regular (valency k), then L(X) is regular (valency 2k-2). 2.If L(X) is regular, then the components of X are regular or bipartite semi-regular.

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Review of topics Definition of line graphs Cliques partitioning edges of L(X) using vertices in X of degree ≥ 2 Beineke’s 9 forbidden subgraphs Non-uniqueness of inverse problem Whitney’s star-triangle result Regularity in X and L(X)

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The End

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