Line graph of a hypergraph: Difference between revisions

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

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

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 setsset]]s. It is a [[generalization]] of the [[line graph]] of a [[Graph (discrete mathematics)|graph]].

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 {{harv|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==

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

{{harvtxt|Beineke|1968}} 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 {{harvtxt|Lovász|1977}} showed there is no such characterization by a finite list if ''k'' = 3.

Beineke<ref>{{harvtxt|KrauszBeineke|19431968}}</ref> characterized line graphs of graphs inby termsa list of 9 [[cliqueforbidden (graphinduced theory)|cliquesubgraph]] coverss. (See [[Linethe graph#Characterizationarticle andon recognition|Line[[line Graphsgraph]]s.) No Acharacterization globalby characterizationforbidden ofinduced Krauszsubgraphs typeis forknown theof line graphs of ''k''-uniform hypergraphs for any ''k'' ≥ 3, was given byand Lovász<ref>{{harvtxt|BergeLovász|19891977}}</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 [[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>

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

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

A global characterization of Krausz type for the line graphs of ''k''-uniform linear hypergraphs for any ''k'' ≥ 3 was given by {{harvtxt|Naik|Rao|Shrikhande|Singhi|1980}}. 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. {{harvtxt|Metelsky|Tyshkevich|1997}} and {{harvtxt|Jacobson|Kézdy|Lehel|1997}} improved this bound to 19. At last {{harvtxt|Skums|Suzdal'|Tyshkevich|2005}} reduced this bound to 16. {{harvtxt|Metelsky|Tyshkevich|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.

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|20052009}}</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.

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

{{harvs|last1 = Naik|last2 = Rao|last3 = Shrikhande|last4 = Singhi|year = 1980|year2 = 1982|txt = yes}} 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 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>

{{harvsharvtxt|last1 = Naik|last2 = Rao|last3 = Shrikhande|last4 = Singhi|year1980}}, = 1980{{harvtxt|Naik|Rao|Shrikhande|Singhi|year2 = 1982|txt = yes}}</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>{{harvsharvtxt|last1 = Naik|last2 = Rao|last3 = Shrikhande|last4 = Singhi|year = 1980|year2}}, = 1982{{harvtxt|nb = yesNaik|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|20052009}}</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''). AnA [[Clique (graph theory)|clique]] in D(''H'') corresponds to an independent set in L(''H''), and vice- versa.

== References ==

{{Reflist}}

*{{citation

| first = L. W. | last = Beineke

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| publisher = Teubner | ___location = Leipzig | pages = 17–23 | year = 1968}}.

*{{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.

*{{citation

| first1 = J. C. | last1 = Bermond

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| first3 = D. | last3 = Sotteau

| title = Line graphs of hypergraphs I

| journal = [[Discrete Mathematics (journal)|Discrete Mathematics]] | volume = 18 | pages = 235–241 | year = 1977

| issue = 3

|mr=0463003 | doi = 10.1016/0012-365X(77)90127-3| url = https://hal.inria.fr/hal-02360671/file/21-BHS77-L%28H%29.pdf

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*{{citation

| first1 = Ranjan N. | last1 = Naik

| first2 = S. B. | last2 = Rao

| first3 = S. S. | last3 = Shrikhande | authorlink3 = S. S. Shrikhande

Line 107 ⟶ 109:

| journal = European Journal of Combinatorics | volume = 3 | pages = 159–172 | year = 1982

| issue = 2

|mr=0670849 | doi=10.1016/s0195-6698(82)80029-2| doi-access = free }}.

*{{citation

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| title = Edge intersection of linear 3-uniform hypergraphs

| journal = Discrete Mathematics

| volume = 309 | pages = 3500–3517 | year = 2009 | doi=10.1016/j.disc.2007.12.082| doi-access = free}}.

*{{citation