Line graph of a hypergraph: Difference between revisions

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A hypergraph is '''linear''' if any two edges have at most one common vertex. Two edges are '''r-intersecting''' if they share at least r common vertices. '''A k-uniform''' hypergraph is a hypegraph with each edge of size k. Note that simple graphs are linear 2-uniform hypergraphs (a simple graph is loopless and contains no multiple edges). The intersection graph of a graph is usually called as Line graph. The characterization of [[Line graph]]s for Graphs was solved by Van Rooij and Wilf and by Beineke. Beineke's (finite) forbidden subgraph characterization immediately implies a polynomial algorithm to recognize line graphs. A characterization of Line graphs in terms of [[Clique]] covers is given by J. Krausz. In Rooij and Wilf's proof, the notion of even and odd triangles was introduced to characterize line graphs. A tringle in a graph G is called ''even'' if every vertex of the graph G is adjecent to either 0 or 2 vertices, otherwise the triangle is called ''odd''.

For larger values of k > 2, there are infinitely many minimal forbidden induced subgraphs. This does not rule out either the existence of polynomial recognition or the possibility of forbidden subgraph characterization (similar to Beineke's) of particular families of graphs. For r=1 there are very interesting results available for k-uniform hypergraphs, k > 2 by various graph theoristsauthors. The difficulty in finding a characterization of r-intersection graphs is twofold. First, there are infinitely many minimal forbidden subgraphs, even for k=3. Second, many authors have suggested that there is no "Krausz-style" characterization in terms fo clique covers, for k > 2.For m > 0, consider a chain of m diamonds (figure 1) such that consecutive diamonds share vertices of degree two. For k > 2, let us add pendent edges at every vertex of degree 2 or 4 is one family of minimal forbidden graphs. Second, many authors have suggested that there is no "Krausz-style" characterization in terms fo clique covers, for k > 2.

Rao, Singhi, Shrikhande proved the surprising result in [7] that there exists a finite family of forbidden graphs for characterizing graphs with minimum degree at least 69 which are intersection graphs of linear 3-uniform hypergraphs. In [4], Jacobson et al. improved the minimum degree condition to 19 and gave a polynomial algorithm to decide whether a graph is a linear intersection graph of 3-uniform hypergraph. The algorithm follows from a simple recursive characterization of graphs of liner Intersection grpahs of k-uniform hypergraphs and relies on the fact that there is a polynomial time recognition algorithm for members of Line graphs of graphs. In [4] Jacboson et al. could not extend finite forbidden subgraph characterization proved for the minimum degree at least 69
to minimum degree 19.
The complexity of recognizing members of intersection graphs of linear 3-uniform hypergraphs without any minimum degree constraint is not known.

In [7], Rao obtained parallel results for any k > 2 under the additional condition that k³ -2k² + 1 is a lower bound on the 'edge-degree of the graph. Define the edge-degree of the edge uv in G as the sum of the degrees of the vertices u and v in G. Both the results in [7] imply polynomial recognition algorithms for garphs under the corresponding minimum degree and minimum edge-degree conditions.

In [6] Metelsky and Tyshkevich, gave the finite forbidden subgraph characterization for linear k3-uniform hypergraphs with minimum degree at least 19 in G anlogousanalogous to [7].

In progress...