Line graph of a hypergraph: Difference between revisions

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{{short description|Generalization of line graphs to hypergraphs}}

==Linear Intersection (Line) graphs of k-uniform [[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 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]] 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''.

In [[graph theory]], particularly in the theory of [[hypergraph]]s, the '''line graph of a hypergraph''' {{mvar|H}}, denoted {{math|L(''H'')}}, is the [[graph (discrete mathematics)|graph]] whose [[Vertex (graph theory)|vertex]] set is the [[Set (mathematics)|set]] of the hyperedges of {{mvar|H}}, with two vertices adjacent in {{math|L(''H'')}} when their corresponding hyperedges have a nonempty [[intersection]] in {{mvar|H}}. In other words, {{math|L(''H'')}} is the [[intersection graph]] of a family of [[finite set]]s. It is a [[generalization]] of the [[line graph]] of a [[Graph (discrete mathematics)|graph]].

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 theorists. 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. Second, many authors have suggested that there is no "Krausz-style" characterization in terms fo clique covers, for k > 2. For k > 2, let us add pendent edges at every vertex of degree 2 or 4 is one family of minimal forbidden graphs for k > 2.

Questions about line graphs of hypergraphs are often generalizations of questions about line graphs of graphs. For instance, a hypergraph whose edges all have size {{mvar|k}} is called {{nowrap|'''{{mvar|k}}-uniform'''}}. (A 2-uniform hypergraph is a graph). In hypergraph theory, it is often natural to require that hypergraphs be {{nowrap|{{mvar|k}}-uniform}}. Every graph is the line graph of some hypergraph, but, given a fixed edge size {{mvar|k}}, not every graph is a line graph of some {{nowrap|{{mvar|k}}-uniform}} hypergraph. A main problem is to characterize those that are, for each {{math|''k'' ≥ 3}}.

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 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. The complexity of recognizing members of intersection graphs of linear 3-uniform hypergraphs without any minimum degree constraint is not known.

A hypergraph is '''linear''' if each pair of hyperedges intersects in at most one vertex. Every graph is the line graph, not only of some hypergraph, but of some linear hypergraph.<ref>{{harv|Berge|1989}}</ref>

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.

==Line graphs of ''k''-uniform hypergraphs, ''k'' ≥ 3==

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

Beineke<ref>{{harvtxt|Beineke|1968}}</ref> characterized line graphs of graphs by a list of 9 [[forbidden induced subgraph]]s. (See the article on [[line graph]]s.) No characterization by forbidden induced subgraphs is known of line graphs of k-uniform hypergraphs for any ''k'' ≥ 3, and Lovász<ref>{{harvtxt|Lovász|1977}}</ref> showed there is no such characterization by a finite list if ''k'' = 3.

==References==

Krausz<ref>{{harvtxt|Krausz|1943}}</ref> characterized line graphs of graphs in terms of [[clique (graph theory)|clique]] covers. (See [[Line graph#Characterization and recognition|Line Graphs]].) A global characterization of Krausz type for the line graphs of ''k''-uniform hypergraphs for any ''k'' ≥ 3 was given by Berge<ref>{{harvtxt|Berge|1989}}</ref>

1. Berge , C., Hypergraphs, Combinatorics of Finite sets. Amsterdam: North-Holland 1989.

2. Bermond, J.C., Heydemann, M.C., Sotteau, D.: Line graphs of hypergraphs I. Discrete Math. 18 235-241 (1977).

==Line graphs of ''k''-uniform linear hypergraphs, ''k'' ≥ 3==

3. Heydemann, M. C., Scotteau, D,: Line graphs of hypergraphs II. Colloq. MAth. Soc. J. Bolyai 18, 567-582 (1976)

4. M S. Jacobson, Andre E. Kezdy, and Jeno Lehel: Recognizing Intersection Graphs of Linear Uniform Hypergraphs. Graphs and Combinatorics (1977) 13: 359-367.

A global characterization of Krausz type for the line graphs of ''k''-uniform linear hypergraphs for any ''k'' ≥ 3 was given by Naik, Rao, Shrikhande, and Singhi.<ref>{{harvtxt|Naik|Rao|Shrikhande|Singhi|1980}}</ref> At the same time, they found a finite list of forbidden induced subgraphs for linear 3-uniform hypergraphs with minimum vertex degree at least 69. Metelsky|l and Tyshkevich<ref>{{harvtxt|Metelsky|Tyshkevich|1997}}</ref> and Jacobson, Kézdy, and Lehel<ref>{{harvtxt|Jacobson|Kézdy|Lehel|1997}}</ref> improved this bound to 19. At last Skums, Suzdal', and Tyshkevich<ref>{{harvtxt|Skums|Suzdal'|Tyshkevich|2009}}</ref> reduced this bound to 16. Metelsky and Tyshkevich<ref>{{harvtxt|Metelsky|Tyshkevich|1997}}</ref> also proved that, if ''k'' > 3, no such finite list exists for linear ''k''-uniform hypergraphs, no matter what lower bound is placed on the degree.

5. J. Krausz, Demonstration nouvelle d'un theorem de Whitney sur les reseaux, Mat. Fiz. Lapok 50 (1943) pp. 75-89

6. Yury Metelsky and Regina Tyshkevich, On Line graphs of Linear 3-uniform Hypergraphs: J. of Graph Theory 25, 243-251 (1997).

The difficulty in finding a characterization of linear ''k''-uniform hypergraphs is due to the fact that there are infinitely many forbidden induced subgraphs. To give examples, for ''m'' > 0, consider a chain of ''m'' [[diamond graph]]s such that the consecutive diamonds share vertices of degree two. For ''k'' ≥ 3, add pendant edges at every vertex of degree 2 or 4 to get one of the families of minimal forbidden subgraphs of Naik, Rao, Shrikhande, and Singhi<ref>

7.

{{harvtxt|Naik|Rao|Shrikhande|Singhi|1980}}, {{harvtxt|Naik|Rao|Shrikhande|Singhi|1982}}</ref> as shown here. This does not rule out either the existence of a polynomial recognition or the possibility of a forbidden induced subgraph characterization similar to Beineke's of line graphs of graphs.

[[Image:Repeated diamond graph.svg|350px|center]]

There are some interesting characterizations available for line graphs of linear ''k''-uniform hypergraphs due to various authors<ref>{{harvtxt|Naik|Rao|Shrikhande|Singhi|1980}}, {{harvtxt|Naik|Rao|Shrikhande|Singhi|1982}}, {{harvnb|Jacobson|Kézdy|Lehel|1997}}, {{harvnb|Metelsky|Tyshkevich|1997}}, and {{harvnb|Zverovich|2004}}</ref> under constraints on the minimum degree or the minimum edge degree of G. Minimum edge degree at least ''k''³-2''k''²+1 in Naik, Rao, Shrikhande, and Singhi<ref>{{harvtxt|Naik|Rao|Shrikhande|Singhi|1980}}</ref> is reduced to 2''k''²-3''k''+1 in Jacobson, Kézdy, and Lehel<ref>{{harvtxt|Jacobson|Kézdy|Lehel|1997}}</ref> and Zverovich<ref>{{harvtxt|Zverovich|2004}}</ref> to characterize line graphs of ''k''-uniform linear hypergraphs for any ''k'' ≥ 3.

The complexity of recognizing line graphs of linear ''k''-uniform hypergraphs without any constraint on minimum degree (or minimum edge-degree) is not known. For ''k'' = 3 and minimum degree at least 19, recognition is possible in polynomial time.<ref>{{harvnb|Jacobson|Kézdy|Lehel|1997}} and {{harvnb|Metelsky|Tyshkevich|1997}}</ref> Skums, Suzdal', and Tyshkevich<ref>{{harvtxt|Skums|Suzdal'|Tyshkevich|2009}}</ref> reduced the minimum degree to 10.

There are many interesting open problems and conjectures in Naik et al., Jacoboson et al., Metelsky et al. and Zverovich.

== Disjointness graph ==

The '''disjointness graph''' of a hypergraph ''H'', denoted D(''H''), is the graph whose vertex set is the set of the hyperedges of ''H'', with two vertices adjacent in D(''H'') when their corresponding hyperedges are ''disjoint'' in ''H''.<ref>{{Cite journal|last=Meshulam|first=Roy|date=2001-01-01|title=The Clique Complex and Hypergraph Matching|journal=Combinatorica|language=en|volume=21|issue=1|pages=89–94|doi=10.1007/s004930170006|s2cid=207006642|issn=1439-6912}}</ref> In other words, D(''H'') is the [[complement graph]] of L(''H''). A [[Clique (graph theory)|clique]] in D(''H'') corresponds to an independent set in L(''H''), and vice versa.

== References ==

{{Reflist}}

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*{{citation|last=Berge|first=C.|title=Hypergraphs: Combinatorics of Finite Sets|year=1989|___location=Amsterdam|publisher=North-Holland|mr=1013569|authorlink=Claude Berge}}. Translated from the French.

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[[Category:Graph families]]

[[Category:Hypergraphs]]