Dually chordal graph

From HandWiki
Short description: Graph whose maximal clique hypergraph is a hypertree

In the mathematical area of graph theory, an undirected graph G is dually chordal if the hypergraph of its maximal cliques is a hypertree. The name comes from the fact that a graph is chordal if and only if the hypergraph of its maximal cliques is the dual of a hypertree. Originally, these graphs were defined by maximum neighborhood orderings and have a variety of different characterizations.[1] Unlike for chordal graphs, the property of being dually chordal is not hereditary, i.e., induced subgraphs of a dually chordal graph are not necessarily dually chordal (hereditarily dually chordal graphs are exactly the strongly chordal graphs), and a dually chordal graph is in general not a perfect graph.

Dually chordal graphs appeared first under the name HT-graphs.[2]

Characterizations

Dually chordal graphs are the clique graphs of chordal graphs,[3] i.e., the intersection graphs of maximal cliques of chordal graphs.

The following properties are equivalent:[4]

  • G has a maximum neighborhood ordering.
  • There is a spanning tree T of G such that any maximal clique of G induces a subtree in T.
  • The closed neighborhood hypergraph N(G) of G is a hypertree.
  • The maximal clique hypergraph of G is a hypertree.
  • G is the 2-section graph of a hypertree.

The condition on the closed neighborhood hypergraph also implies that a graph is dually chordal if and only if its square is chordal and its closed neighborhood hypergraph has the Helly property.

In (De Caria Gutierrez) dually chordal graphs are characterized in terms of separator properties. In (Brešar 2003) it was shown that dually chordal graphs are precisely the intersection graphs of maximal hypercubes of graphs of acyclic cubical complexes.

The structure and algorithmic use of doubly chordal graphs is given by (Moscarini 1993). These are graphs which are chordal and dually chordal.

Recognition

Dually chordal graphs can be recognized in linear time, and a maximum neighborhood ordering of a dually chordal graph can be found in linear time.[5]

Complexity of problems

While some basic problems such as maximum independent set, maximum clique, coloring and clique cover remain NP-complete for dually chordal graphs, some variants of the minimum dominating set problem and Steiner tree are efficiently solvable on dually chordal graphs (but Independent Domination remains NP-complete).[6] See (Brandstädt Chepoi) for the use of dually chordal graph properties for tree spanners, and see (Brandstädt Leitert) for a linear time algorithm of efficient domination and efficient edge domination on dually chordal graphs.

Notes

  1. ↑ (Brandstädt Dragan); (Brandstädt Le)
  2. ↑ (Dragan 1989); (Dragan Prisacaru); (Dragan 1993)
  3. ↑ (Gutierrez Oubina); (Szwarcfiter Bornstein)
  4. ↑ (Brandstädt Chepoi);(Brandstädt Dragan); (Dragan Prisacaru); (Dragan 1993)
  5. ↑ (Brandstädt Chepoi);(Dragan 1989)
  6. ↑ (Brandstädt Chepoi); (Dragan Prisacaru); (Dragan 1993)

References

  • Brandstädt, Andreas; Chepoi, Victor; Dragan, Feodor (1998), "The algorithmic use of hypertree structure and maximum neighborhood orderings", Discrete Applied Mathematics 82 (1–3): 43–77, doi:10.1016/s0166-218x(97)00125-x .
  • Brandstädt, Andreas; Chepoi, Victor; Dragan, Feodor (1999), "Distance approximating trees for chordal and dually chordal graphs", Journal of Algorithms 30: 166–184, doi:10.1006/jagm.1998.0962 .
  • Brandstädt, Andreas; Dragan, Feodor; Chepoi, Victor; Voloshin, Vitaly (1998), "Dually chordal graphs", SIAM Journal on Discrete Mathematics 11 (3): 437–455, doi:10.1137/S0895480193253415 .
  • Brandstädt, Andreas; Le, Van Bang; Spinrad, Jeremy (1999), Graph Classes: A Survey, SIAM Monographs on Discrete Mathematics and Applications, ISBN 0-89871-432-X, https://archive.org/details/graphclassessurv0000bran .
  • Brandstädt, Andreas; Leitert, Arne; Rautenbach, Dieter (2012), "Efficient dominating and edge dominating sets for graphs and hypergraphs", in Chao, Kun-Mao; Hsu, Tsan-sheng; Lee, Der-Tsai, Algorithms and Computation – 23rd International Symposium, ISAAC 2012, Taipei, Taiwan, December 19–21, 2012. Proceedings, Lecture Notes in Computer Science, 7676, Springer, pp. 267–277, doi:10.1007/978-3-642-35261-4_30 .
  • Brešar, Boštjan (2003), "Intersection graphs of maximal hypercubes", European Journal of Combinatorics 24 (2): 195–209, doi:10.1016/s0195-6698(02)00142-7 .
  • De Caria, Pablo; Gutierrez, Marisa (2012), "On Minimal Vertex Separators of Dually Chordal Graphs: Properties and Characterizations", Discrete Applied Mathematics 160 (18): 2627–2635, doi:10.1016/j.dam.2012.02.022 .
  • Dragan, Feodor (1989), Centers of Graphs and the Helly Property (in Russian), Ph.D. thesis, Moldova State University, Moldova .
  • Dragan, Feodor; Prisacaru, Chiril; Chepoi, Victor (1992), "Location problems in graphs and the Helly property (in Russian)", Discrete Math. (Moscow) 4: 67–73 .
  • Dragan, Feodor (1993), "HT-graphs: centers, connected r-domination and Steiner trees", Comput. Sci. J. of Moldova (Kishinev) 1: 64–83 .
  • Gutierrez, Marisa; Oubina, L. (1996), "Metric Characterizations of proper Interval Graphs and Tree-Clique Graphs", Journal of Graph Theory 21 (2): 199–205, doi:10.1002/(sici)1097-0118(199602)21:2<199::aid-jgt9>3.0.co;2-m .
  • McKee, Terry A.; McMorris, FR. (1999), Topics in Intersection Graph Theory, SIAM Monographs on Discrete Mathematics and Applications .
  • Moscarini, Marina (1993), "Doubly Chordal Graphs, Steiner trees and connected domination", Networks 23: 59–69, doi:10.1002/net.3230230108 .
  • Szwarcfiter, Jayme L.; Bornstein, Claudson F. (1994), "Clique Graphs of Chordal and Path Graphs", SIAM Journal on Discrete Mathematics 7 (2): 331–336, doi:10.1137/s0895480191223191 .



Retrieved from "https://handwiki.org/wiki/index.php?title=Dually_chordal_graph&oldid=3368242"