, Volume 80, Issue 10, pp 2834–2848 | Cite as

The Induced Separation Dimension of a Graph

  • Emile Ziedan
  • Deepak RajendraprasadEmail author
  • Rogers Mathew
  • Martin Charles Golumbic
  • Jérémie Dusart


A linear ordering of the vertices of a graph G separates two edges of G if both the endpoints of one precede both the endpoints of the other in the order. We call two edges \(\{a,b\}\) and \(\{c,d\}\) of G strongly independent if the set of endpoints \(\{a,b,c,d\}\) induces a \(2K_2\) in G. The induced separation dimension of a graph G is the smallest cardinality of a family \(\mathcal {L}\) of linear orders of V(G) such that every pair of strongly independent edges in G are separated in at least one of the linear orders in \(\mathcal {L}\). For each \(k \in \mathbb {N}\), the family of graphs with induced separation dimension at most k is denoted by \({\text {ISD}}(k)\). In this article, we initiate a study of this new dimensional parameter. The class \({\text {ISD}}(1)\) or, equivalently, the family of graphs which can be embedded on a line so that every pair of strongly independent edges are disjoint line segments, is already an interesting case. On the positive side, we give characterizations for chordal graphs in \({\text {ISD}}(1)\) which immediately lead to a polynomial time algorithm which determines the induced separation dimension of chordal graphs. On the negative side, we show that the recognition problem for \({\text {ISD}}(1)\) is NP-complete for general graphs. Nevertheless, we show that the maximum induced matching problem can be solved efficiently in \({\text {ISD}}(1)\). We then briefly study \({\text {ISD}}(2)\) and show that it contains many important graph classes like outerplanar graphs, chordal graphs, circular arc graphs and polygon-circle graphs. Finally, we describe two techniques to construct graphs with large induced separation dimension. The first one is used to show that the maximum induced separation dimension of a graph on n vertices is \(\Theta (\lg n)\) and the second one is used to construct AT-free graphs with arbitrarily large induced separation dimension. The second construction is also used to show that, for every \(k \ge 2\), the recognition problem for \({\text {ISD}}(k)\) is NP-complete even on AT-free graphs.


Induced separation dimension Vertex ordering Acyclic orientation Asteroidal triples 

Mathematics Subject Classification

05C75 05C62 05C85 


  1. 1.
    Alon, N., Basavaraju, M., Chandran, L.S., Mathew, R., Rajendraprasad, D.: Separation dimension of bounded degree graphs. SIAM J. Discrete Math. 29(1), 59–64 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Basavaraju, M., Chandran, L.S., Golumbic, M.C., Mathew, R., Rajendraprasad, D.: Boxicity and separation dimension. In: Graph-Theoretic Concepts in Computer Science : Proceedings of the 40th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2014. LNCS, vol. 8747, pp. 81–92. Springer International Publishing, (2014)Google Scholar
  3. 3.
    Basavaraju, M., Chandran, L.S., Golumbic, M.C., Mathew, R., Rajendraprasad, D.: Separation dimension of graphs and hypergraphs. Algorithmica 75, 187–204 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph Classes: A Survey. SIAM, Philadelphia (1999)CrossRefzbMATHGoogle Scholar
  5. 5.
    Broersma, H., Patel, V., Pyatkin, A.: On toughness and Hamiltonicity of \(2K_2\)-free graphs. J. Graph Theory 75(3), 244–255 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cameron, K.: Induced matchings. Discrete Appl. Math. 24(1), 97–102 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cameron, K.: Induced matchings in intersection graphs. Discrete Math. 278(1), 1–9 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cameron, K., Sritharan, R., Tang, Y.: Finding a maximum induced matching in weakly chordal graphs. Discrete Math. 266(1), 133–142 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chang, J.-M.: Induced matchings in asteroidal triple-free graphs. Discrete Appl. Math. 132(1), 67–78 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Chung, F.R.K., Gyárfás, A., Tuza, Z., Trotter, W.T.: The maximum number of edges in \(2K_2\)-free graphs of bounded degree. Discrete Math. 81(2), 129–135 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Chvátal, V., Hammer, P.L.: Aggregation of inequalities in integer programming. Ann. Discrete Math. 1, 145–162 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Corneil, D.G., Stacho, J.: Vertex ordering characterizations of graphs of bounded asteroidal number. J. Graph Theory 78(1), 61–79 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Eaton, N., Faubert, G.: Caterpillar tolerance representations. Bull. Inst. Comb. Appl. 64, 109–117 (2012)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Fishburn, P.C., Trotter, W.T.: Dimensions of hypergraphs. J. Comb. Theory Ser. B 56(2), 278–295 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Gavril, F.: Maximum weight independent sets and cliques in intersection graphs of filaments. Inf. Process. Lett. 73(5), 181–188 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Elsevier, Amsterdam (2004)zbMATHGoogle Scholar
  17. 17.
    Golumbic, M.C., Lewenstein, M.: New results on induced matchings. Discrete Appl. Math. 101(1), 157–165 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Golumbic, M.C., Trenk, A.N.: Tolerance Graphs. Cambridge University Press, Cambridge (2004)CrossRefzbMATHGoogle Scholar
  19. 19.
    Habib, M., Paul, C., Telle, J.A.: A linear-time algorithm for recognition of catval graphs. In: Eurocomb 2003: European Conference on Combinatorics, Graphs Theory and Applications (2003)Google Scholar
  20. 20.
    Lovász, L: Coverings and colorings of hypergraphs. In: Proceedings of 4th Southeastern Conference on Combinatorics, Graph Theory, and Computing, pp. 3–12. Utilitas Mathematica Publishing, Winnipeg (1973)Google Scholar
  21. 21.
    Lozin, V.V., Kaminski, M.: Coloring edges and vertices of graphs without short or long cycles. Contrib. Discrete Math. 2(1), 61–66 (2007)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Lozin, V.V., Mosca, R.: Independent sets in extensions of \(2K_2\)-free graphs. Discrete Appl. Math. 146(1), 74–80 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Mycielski, J.: Sur le coloriage des graphes. Colloq. Math. 3, 161–162 (1955)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Roberts, F.S.: On the boxicity and cubicity of a graph. In: Tutte, W.T. (ed.) Recent Progresses in Combinatorics, pp. 301–310. Academic Press, New York (1969)Google Scholar
  25. 25.
    Telle, J.A.: Tree-decompositions of small pathwidth. Discrete Appl. Math. 145(2), 210–218 (2005)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of Computer Science, The Caesarea Rothschild InstituteUniversity of HaifaHaifaIsrael
  2. 2.Department of Computer ScienceIndian Institute of TechnologyKharagpurIndia

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