Line graph of a hypergraph: Difference between revisions

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==Linear Intersection (Line) graphs of k-uniform linear [[Hypergraph]] ==

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 linear 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''.

Let L^k₁ stand for the family of intersection graphs of k-uniform linear hypergraphs. Let d(G) denote the minimum degree of the graph G. 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 authors. The difficulty in finding a characterization of r-intersection graphs is twofold. First, there are infinitely many minimal forbidden subgraphs, even for k=3. 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