Line graph of a hypergraph: Difference between revisions

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

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In [[mathematics]], a [[hypergraph]] is a generalization of a [[graph (mathematics)|graph]], where edges can contain any finite number of vertices, and the line graph of a hypergraph is a generalization of the [[line graph]] of a graph. The precise definition is that the '''line graph''' of a hypergraph is the graph whose vertex set is the set of edges of the hypergraph, with two edges adjacent when they have nonempty intersection. Thus, the line graph of a hypergraph is the same as the [[intersection graph]] of a family of finite sets.

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]].

However, the questions asked tend to be different. 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 ''k'' is called ''k''' ''-uniform'''. (A 2-uniform hypergraph is a graph.). In hypergraph theory, it is often natural to require that hypergraphs be ''k''-uniform. Every graph is the line graph of some hypergraph, but, given a fixed edge size ''k'', not every graph is a line graph of some ''k''-uniform hypergraph. A main problem is to characterize those that are, for each ''k'' ≥ 3.

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}}.

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 (Berge 1989).

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>

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

Beineke (<ref>{{harvtxt|Beineke|1968)}}</ref> characterized line graphs of graphs by a list of 9 [[forbidden induced subgraphssubgraph]]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 Lóvász (Lovász<ref>{{harvtxt|Lovász|1977)}}</ref> showed there is no such characterization by a finite list if ''k'' = 3.

Krausz (<ref>{{harvtxt|Krausz|1943)}}</ref> characterized line graphs of graphs in terms of [[clique (graph theory)|clique]] covers. (See the[[Line articlegraph#Characterization onand linerecognition|Line graphsGraphs]].) 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>

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

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.

==Line graphs of ''k''-uniform linear hypergraphs, ''k'' &ge; 3==

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>

A global characterization of Krausz type for the line graphs of ''k''-uniform linear hypergraphs for any ''k'' ≥ 3 was given by Naik et al. (1980).

{{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]]

Naik et al. (1980) found a finite list of forbidden induced subgraphs for linear 3-uniform hypergraphs with minimum vertex degree at least 69. Metelsky et al. (1997) and Jacobson et al.(1997) improved this bound to 19. Metelsky et al. (1997) 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.

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 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 graphs 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 in Naik et al. (1980, 1982), 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.

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.

[[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 (Naik et al. 1980, 1982, Jacobson et al. 1997, Metelsky et al. 1997, and Zverovich 2004.) 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 et al. is reduced to 2''k''²-3''k''+1 in Jacobson et al. (1997) and Zverovich (2004) 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 (Metelsky et al. 1997 and Jacobson et al. 1997). Skums et al. (2005) 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 ==

==References==

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.

* L. W. Beineke (1968), On derived graphs and digraphs. In: Beitrage zur Graphentheorie [H. Saks et al., eds.], Teubner, Leipzig, pp. 17-23.

* C. Berge, 1989, ''Hypergraphs: Combinatorics of Finite Sets''. Amsterdam: North-Holland.

== References ==

* J.C. Bermond, M.C. Heydemann, and D. Sotteau (1977), Line graphs of hypergraphs I. ''Discrete Mathematics'', vol. 18, pp. 235-241.

{{Reflist}}

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[[Category:Graph families]]

[[Category:Hypergraphs]]