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Line graph of a hypergraph

November 24, 2023

In graph theory, particularly in the theory of hypergraphs, the line graph of a hypergraph H, denoted L(H), is the 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.

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.

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.[1]

Line graphs of k-uniform hypergraphs, k ≥ 3 edit

Beineke[2] characterized line graphs of graphs by a list of 9 forbidden induced subgraphs. (See the article on line graphs.) No characterization by forbidden induced subgraphs is known of line graphs of k-uniform hypergraphs for any k ≥ 3, and Lovász[3] showed there is no such characterization by a finite list if k = 3.

Krausz[4] characterized line graphs of graphs in terms of clique covers. (See Line Graphs.) A global characterization of Krausz type for the line graphs of k-uniform hypergraphs for any k ≥ 3 was given by Berge[5]

Line graphs of k-uniform linear hypergraphs, k ≥ 3 edit

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.[6] 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[7] and Jacobson, Kézdy, and Lehel[8] improved this bound to 19. At last Skums, Suzdal', and Tyshkevich[9] reduced this bound to 16. Metelsky and Tyshkevich[10] 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 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 of Naik, Rao, Shrikhande, and Singhi[11] 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.

 

There are some interesting characterizations available for line graphs of linear k-uniform hypergraphs due to various authors[12] under constraints on the minimum degree or the minimum edge degree of G. Minimum edge degree at least k3-2k2+1 in Naik, Rao, Shrikhande, and Singhi[13] is reduced to 2k2-3k+1 in Jacobson, Kézdy, and Lehel[14] and Zverovich[15] 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.[16] Skums, Suzdal', and Tyshkevich[17] 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 edit

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.[18] In other words, D(H) is the complement graph of L(H). A clique in D(H) corresponds to an independent set in L(H), and vice versa.

References edit

  1. ^ (Berge 1989)
  2. ^ Beineke (1968)
  3. ^ Lovász (1977)
  4. ^ Krausz (1943)
  5. ^ Berge (1989)
  6. ^ Naik et al. (1980)
  7. ^ Metelsky & Tyshkevich (1997)
  8. ^ Jacobson, Kézdy & Lehel (1997)
  9. ^ Skums, Suzdal' & Tyshkevich (2009)
  10. ^ Metelsky & Tyshkevich (1997)
  11. ^ Naik et al. (1980), Naik et al. (1982)
  12. ^ Naik et al. (1980), Naik et al. (1982), Jacobson, Kézdy & Lehel 1997, Metelsky & Tyshkevich 1997, and Zverovich 2004
  13. ^ Naik et al. (1980)
  14. ^ Jacobson, Kézdy & Lehel (1997)
  15. ^ Zverovich (2004)
  16. ^ Jacobson, Kézdy & Lehel 1997 and Metelsky & Tyshkevich 1997
  17. ^ Skums, Suzdal' & Tyshkevich (2009)
  18. ^ Meshulam, Roy (2001-01-01). "The Clique Complex and Hypergraph Matching". Combinatorica. 21 (1): 89–94. doi:10.1007/s004930170006. ISSN 1439-6912. S2CID 207006642.
  • Beineke, L. W. (1968), "On derived graphs and digraphs", in Sachs, H.; Voss, H.; Walther, H. (eds.), Beitrage zur Graphentheorie, Leipzig: Teubner, pp. 17–23.
  • Berge, C. (1989), Hypergraphs: Combinatorics of Finite Sets, Amsterdam: North-Holland, MR 1013569. Translated from the French.
  • Bermond, J. C.; Heydemann, M. C.; Sotteau, D. (1977), "Line graphs of hypergraphs I" (PDF), Discrete Mathematics, 18 (3): 235–241, doi:10.1016/0012-365X(77)90127-3, MR 0463003.
  • Heydemann, M. C.; Sotteau, D. (1976), "Line graphs of hypergraphs II", Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely, 1976), Colloq. Math. Soc. J. Bolyai, vol. 18, pp. 567–582, MR 0519291.
  • Krausz, J. (1943), "Démonstration nouvelle d'une théorème de Whitney sur les réseaux", Mat. Fiz. Lapok, 50: 75–85, MR 0018403. (In Hungarian, with French abstract.)
  • Lovász, L. (1977), "Problem 9", Beiträge zur Graphentheorie und deren Anwendungen, Vorgetragen auf dem Internationalen Kolloquium in Oberhof (DDR), p. 313.
  • Jacobson, M. S.; Kézdy, Andre E.; Lehel, Jeno (1997), "Recognizing intersection graphs of linear uniform hypergraphs", Graphs and Combinatorics, 13 (4): 359–367, doi:10.1007/BF03353014, MR 1485929, S2CID 9173731.
  • Metelsky, Yury; Tyshkevich, Regina (1997), "On line graphs of linear 3-uniform hypergraphs", Journal of Graph Theory, 25 (4): 243–251, doi:10.1002/(SICI)1097-0118(199708)25:4<243::AID-JGT1>3.0.CO;2-K, MR 1459889.
  • Naik, Ranjan N.; Rao, S. B.; Shrikhande, S. S.; Singhi, N. M. (1980), "Intersection graphs of k-uniform hypergraphs", Combinatorial mathematics, optimal designs and their applications (Proc. Sympos. Combin. Math. and Optimal Design, Colorado State Univ., Fort Collins, Colo., 1978), Annals of Discrete Mathematics, vol. 6, pp. 275–279, MR 0593539.
  • Naik, Ranjan N.; Rao, S. B.; Shrikhande, S. S.; Singhi, N. M. (1982), "Intersection graphs of k-uniform linear hypergraphs", European Journal of Combinatorics, 3 (2): 159–172, doi:10.1016/s0195-6698(82)80029-2, MR 0670849.
  • Skums, P. V.; Suzdal', S. V.; Tyshkevich, R. I. (2009), "Edge intersection of linear 3-uniform hypergraphs", Discrete Mathematics, 309: 3500–3517, doi:10.1016/j.disc.2007.12.082.
  • Zverovich, Igor E. (2004), "A solution to a problem of Jacobson, Kézdy and Lehel", Graphs and Combinatorics, 20 (4): 571–577, doi:10.1007/s00373-004-0572-1, MR 2108401, S2CID 33662052.

November 24, 2023
line, graph, hypergraph, graph, theory, particularly, theory, hypergraphs, line, graph, hypergraph, denoted, graph, whose, vertex, hyperedges, with, vertices, adjacent, when, their, corresponding, hyperedges, have, nonempty, intersection, other, words, interse. In graph theory particularly in the theory of hypergraphs the line graph of a hypergraph H denoted L H is the 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 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 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 1 Contents 1 Line graphs of k uniform hypergraphs k 3 2 Line graphs of k uniform linear hypergraphs k 3 3 Disjointness graph 4 ReferencesLine graphs of k uniform hypergraphs k 3 editBeineke 2 characterized line graphs of graphs by a list of 9 forbidden induced subgraphs See the article on line graphs No characterization by forbidden induced subgraphs is known of line graphs of k uniform hypergraphs for any k 3 and Lovasz 3 showed there is no such characterization by a finite list if k 3 Krausz 4 characterized line graphs of graphs in terms of clique covers See Line Graphs A global characterization of Krausz type for the line graphs of k uniform hypergraphs for any k 3 was given by Berge 5 Line graphs of k uniform linear hypergraphs k 3 editA 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 6 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 7 and Jacobson Kezdy and Lehel 8 improved this bound to 19 At last Skums Suzdal and Tyshkevich 9 reduced this bound to 16 Metelsky and Tyshkevich 10 also proved that if k gt 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 gt 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 of Naik Rao Shrikhande and Singhi 11 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 nbsp There are some interesting characterizations available for line graphs of linear k uniform hypergraphs due to various authors 12 under constraints on the minimum degree or the minimum edge degree of G Minimum edge degree at least k3 2k2 1 in Naik Rao Shrikhande and Singhi 13 is reduced to 2k2 3k 1 in Jacobson Kezdy and Lehel 14 and Zverovich 15 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 16 Skums Suzdal and Tyshkevich 17 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 editThe 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 18 In other words D H is the complement graph of L H A clique in D H corresponds to an independent set in L H and vice versa References edit Berge 1989 Beineke 1968 Lovasz 1977 Krausz 1943 Berge 1989 Naik et al 1980 Metelsky amp Tyshkevich 1997 Jacobson Kezdy amp Lehel 1997 Skums Suzdal amp Tyshkevich 2009 Metelsky amp Tyshkevich 1997 Naik et al 1980 Naik et al 1982 Naik et al 1980 Naik et al 1982 Jacobson Kezdy amp Lehel 1997 Metelsky amp Tyshkevich 1997 and Zverovich 2004 Naik et al 1980 Jacobson Kezdy amp Lehel 1997 Zverovich 2004 Jacobson Kezdy amp Lehel 1997 and Metelsky amp Tyshkevich 1997 Skums Suzdal amp Tyshkevich 2009 Meshulam Roy 2001 01 01 The Clique Complex and Hypergraph Matching Combinatorica 21 1 89 94 doi 10 1007 s004930170006 ISSN 1439 6912 S2CID 207006642 Beineke L W 1968 On derived graphs and digraphs in Sachs H Voss H Walther H eds Beitrage zur Graphentheorie Leipzig Teubner pp 17 23 Berge C 1989 Hypergraphs Combinatorics of Finite Sets Amsterdam North Holland MR 1013569 Translated from the French Bermond J C Heydemann M C Sotteau D 1977 Line graphs of hypergraphs I PDF Discrete Mathematics 18 3 235 241 doi 10 1016 0012 365X 77 90127 3 MR 0463003 Heydemann M C Sotteau D 1976 Line graphs of hypergraphs II Combinatorics Proc Fifth Hungarian Colloq Keszthely 1976 Colloq Math Soc J Bolyai vol 18 pp 567 582 MR 0519291 Krausz J 1943 Demonstration nouvelle d une theoreme de Whitney sur les reseaux Mat Fiz Lapok 50 75 85 MR 0018403 In Hungarian with French abstract Lovasz L 1977 Problem 9 Beitrage zur Graphentheorie und deren Anwendungen Vorgetragen auf dem Internationalen Kolloquium in Oberhof DDR p 313 Jacobson M S Kezdy Andre E Lehel Jeno 1997 Recognizing intersection graphs of linear uniform hypergraphs Graphs and Combinatorics 13 4 359 367 doi 10 1007 BF03353014 MR 1485929 S2CID 9173731 Metelsky Yury Tyshkevich Regina 1997 On line graphs of linear 3 uniform hypergraphs Journal of Graph Theory 25 4 243 251 doi 10 1002 SICI 1097 0118 199708 25 4 lt 243 AID JGT1 gt 3 0 CO 2 K MR 1459889 Naik Ranjan N Rao S B Shrikhande S S Singhi N M 1980 Intersection graphs of k uniform hypergraphs Combinatorial mathematics optimal designs and their applications Proc Sympos Combin Math and Optimal Design Colorado State Univ Fort Collins Colo 1978 Annals of Discrete Mathematics vol 6 pp 275 279 MR 0593539 Naik Ranjan N Rao S B Shrikhande S S Singhi N M 1982 Intersection graphs of k uniform linear hypergraphs European Journal of Combinatorics 3 2 159 172 doi 10 1016 s0195 6698 82 80029 2 MR 0670849 Skums P V Suzdal S V Tyshkevich R I 2009 Edge intersection of linear 3 uniform hypergraphs Discrete Mathematics 309 3500 3517 doi 10 1016 j disc 2007 12 082 Zverovich Igor E 2004 A solution to a problem of Jacobson Kezdy and Lehel Graphs and Combinatorics 20 4 571 577 doi 10 1007 s00373 004 0572 1 MR 2108401 S2CID 33662052 Voloshin Vitaly I 2009 Introduction to Graph and Hypergraph Theory New York Nova Science Publishers Inc MR 2514872 Retrieved from https en wikipedia org w index php title Line graph of a hypergraph amp oldid 1184477359, wikipedia, wiki, book, books, library,

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