Skip to main content
SpringerLink
Log in

On Local Structures of Cubicity 2 Graphs

  • Conference paper
  • First Online:
Combinatorial Optimization and Applications (COCOA 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10043))

Abstract

A 2-stab unit interval graph (2SUIG) is an axes-parallel unit square intersection graph where the unit squares intersect either of the two fixed lines parallel to the X-axis, distance \(1 + \epsilon \) (\(0< \epsilon < 1\)) apart. This family of graphs allow us to study local structures of unit square intersection graphs, that is, graphs with cubicity 2. The complexity of determining whether a tree has cubicity 2 is unknown while the graph recognition problem for unit square intersection graph is known to be NP-hard. We present a linear time algorithm for recognizing trees that admit a 2SUIG representation.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Bhatt, S.N., Cosmadakis, S.S.: The complexity of minimizing wire lengths in VLSI layouts. Inf. Process. Lett. 25(4), 263–267 (1987)

    Article  MATH  Google Scholar 

  2. Bhore, S.K., Chakraborty, D., Das, S., Sen, S.: On a special class of boxicity 2 graphs. In: Ganguly, S., Krishnamurti, R. (eds.) CALDAM 2015. LNCS, vol. 8959, pp. 157–168. Springer, Heidelberg (2015)

    Google Scholar 

  3. Breu, H.: Algorithmic aspects of constrained unit disk graphs. Ph.D. thesis, University of British Columbia (1996)

    Google Scholar 

  4. Clark, B.N., Colbourn, C.J., Johnson, D.S.: Unit disk graphs. Discrete Math. 86(1–3), 165–177 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  5. Correa, J., Feuilloley, L., Pérez-Lantero, P., Soto, J.A.: Independent and hitting sets of rectangles intersecting a diagonal line: algorithms and complexity. Discrete Comput. Geom. 53(2), 344–365 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. Imai, H., Asano, T.: Finding the connected components and a maximum clique of an intersection graph of rectangles in the plane. J. Algorithms 4(4), 310–323 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  7. Babu, J., Basavaraju, M., Chandran, L.S., Rajendraprasad, D., Sivadasan, N.: Approximating the cubicity of trees. CoRR, abs/1402.6310 (2014)

    Google Scholar 

  8. Kratochvíl, J.: A special planar satisfiability problem and a consequence of its NP-completeness. Discrete Appl. Math. 52(3), 233–252 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  9. Lekkeikerker, C., Boland, J.: Representation of a finite graph by a set of intervals on the real line. Fundamenta Math. 51(1), 45–64 (1962)

    MathSciNet  Google Scholar 

  10. Roberts, F.S.: On the boxicity and cubicity of a graph. In: Recent Progresses in Combinatorics, pp. 301–310 (1969)

    Google Scholar 

  11. Scheinerman, E.R.: Characterizing intersection classes of graphs. Discrete Math. 55(2), 185–193 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  12. West, D.B., Shmoys, D.B.: Recognizing graphs with fixed interval number is NP-complete. Discrete Appl. Math. 8(3), 295–305 (1984)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Indian Statistical Institute, Kolkata, India

    Dibyayan Chakraborty & Sandip Das

  2. Ben-Gurion University, Beersheba, Israel

    Sujoy Bhore

  3. Indian Statistical Institute, Bangalore, India

    Sagnik Sen

Authors
  1. Sujoy Bhore
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dibyayan Chakraborty
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sandip Das
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Sagnik Sen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dibyayan Chakraborty .

Editor information

Editors and Affiliations

  1. University of Hong Kong, Hong Kong, China

    T-H. Hubert Chan

  2. City University of Hong Kong, Hong Kong, China

    Minming Li

  3. City University of Hong Kong, Hong Kong, China

    Lusheng Wang

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing AG

About this paper

Cite this paper

Bhore, S., Chakraborty, D., Das, S., Sen, S. (2016). On Local Structures of Cubicity 2 Graphs. In: Chan, TH., Li, M., Wang, L. (eds) Combinatorial Optimization and Applications. COCOA 2016. Lecture Notes in Computer Science(), vol 10043. Springer, Cham. https://doi.org/10.1007/978-3-319-48749-6_19

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-48749-6_19

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-48748-9

  • Online ISBN: 978-3-319-48749-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation