Minimal rankings of the Cartesian product Kₙ ☐ Kₘ

Gilbert Eyabi; Jobby Jacob; Renu C. Laskar; Darren A. Narayan; Dan Pillone

Discussiones Mathematicae Graph Theory (2012)

  • Volume: 32, Issue: 4, page 725-735
  • ISSN: 2083-5892

Abstract

top
For a graph G = (V, E), a function f:V(G) → 1,2, ...,k is a k-ranking if f(u) = f(v) implies that every u - v path contains a vertex w such that f(w) > f(u). A k-ranking is minimal if decreasing any label violates the definition of ranking. The arank number, ψ r ( G ) , of G is the maximum value of k such that G has a minimal k-ranking. We completely determine the arank number of the Cartesian product Kₙ ☐ Kₙ, and we investigate the arank number of Kₙ ☐ Kₘ where n > m.

How to cite

top

Gilbert Eyabi, et al. "Minimal rankings of the Cartesian product Kₙ ☐ Kₘ." Discussiones Mathematicae Graph Theory 32.4 (2012): 725-735. <http://eudml.org/doc/270880>.

@article{GilbertEyabi2012,
abstract = {For a graph G = (V, E), a function f:V(G) → 1,2, ...,k is a k-ranking if f(u) = f(v) implies that every u - v path contains a vertex w such that f(w) > f(u). A k-ranking is minimal if decreasing any label violates the definition of ranking. The arank number, $ψ_r(G)$, of G is the maximum value of k such that G has a minimal k-ranking. We completely determine the arank number of the Cartesian product Kₙ ☐ Kₙ, and we investigate the arank number of Kₙ ☐ Kₘ where n > m.},
author = {Gilbert Eyabi, Jobby Jacob, Renu C. Laskar, Darren A. Narayan, Dan Pillone},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {graph colorings; rankings of graphs; minimal rankings; rank number; arank number; Cartesian product of graphs; rook's graph},
language = {eng},
number = {4},
pages = {725-735},
title = {Minimal rankings of the Cartesian product Kₙ ☐ Kₘ},
url = {http://eudml.org/doc/270880},
volume = {32},
year = {2012},
}

TY - JOUR
AU - Gilbert Eyabi
AU - Jobby Jacob
AU - Renu C. Laskar
AU - Darren A. Narayan
AU - Dan Pillone
TI - Minimal rankings of the Cartesian product Kₙ ☐ Kₘ
JO - Discussiones Mathematicae Graph Theory
PY - 2012
VL - 32
IS - 4
SP - 725
EP - 735
AB - For a graph G = (V, E), a function f:V(G) → 1,2, ...,k is a k-ranking if f(u) = f(v) implies that every u - v path contains a vertex w such that f(w) > f(u). A k-ranking is minimal if decreasing any label violates the definition of ranking. The arank number, $ψ_r(G)$, of G is the maximum value of k such that G has a minimal k-ranking. We completely determine the arank number of the Cartesian product Kₙ ☐ Kₙ, and we investigate the arank number of Kₙ ☐ Kₘ where n > m.
LA - eng
KW - graph colorings; rankings of graphs; minimal rankings; rank number; arank number; Cartesian product of graphs; rook's graph
UR - http://eudml.org/doc/270880
ER -

References

top
  1. [1] H.L. Bodlaender, J.S. Deogun, K. Jansen, T. Kloks, D. Kratsch, H. Müller and Zs. Tuza, Rankings of graphs, Graph-theoretic concepts in computer science (Herrsching, 1994), Lect. Notes Comput. Sci. 903 (1995) 292-304. 
  2. [2] P.De la Torre, R. Greenlaw and A. Schaeffer, Optimal ranking of trees in polynomial time, In: Proc. 4^{th} ACM Symp. on Discrete Algorithms, Austin Texas, (1993) 138-144. Zbl0801.68129
  3. [3] I.S. Duff and J.K. Reid, The multifrontal solution of indefinite sparse symmetric linear equations, ACM Trans. Math. Software (1983) 9 302-325, doi: 10.1145/356044.356047. Zbl0515.65022
  4. [4] J. Ghoshal, R. Laskar and D. Pillone, Minimal rankings, Networks 28 ( 1996) 45-53, doi: 10.1002/(SICI)1097-0037(199608)28:1<45::AID-NET6>3.0.CO;2-D Zbl0863.05071
  5. [5] J. Ghoshal, R. Laskar and D. Pillone, Further results on minimal rankings, Ars Combin. 52 (1999) 181-198. Zbl0977.05048
  6. [6] S. Hsieh, On vertex ranking of a starlike graph, Inform. Process. Lett. 82 (2002) 131-135, doi: 10.1016/S0020-0190(01)00262-9. 
  7. [7] A.V. Iyer, H.D. Ratliff and G. Vijayan, Optimal node ranking of trees, Inform. Process. Lett. 28 (1988) 225-229, doi: 10.1016/0020-0190(88)90194-9. Zbl0661.68063
  8. [8] V. Kostyuk, D.A. Narayan and V.A. Williams, Minimal rankings and the arank number of a path, Discrete Math. 306 (2006) 1991-1996, doi: 10.1016/j.disc.2006.01.027. Zbl1101.05040
  9. [9] R. Laskar and D. Pillone, Theoretical and complexity results for minimal rankings, J. Combin. Inform. System Sci. 25) (2000) 17-33. Zbl1219.05188
  10. [10] R. Laskar and D. Pillone, Extremal results in rankings, Congr. Numer. 149 (2001) 33-54. Zbl0989.05058
  11. [11] C. Leiserson, Area efficient graph layouts for VLSI, Proc. 21st Ann IEEE Symp. FOCS (1980) 270-281. 
  12. [12] J.W.H. Liu, The role of elimination trees in sparse factorization, SIAM J. Matrix Anal. Appl. 11 (1990) 134-172, doi: 10.1137/0611010. Zbl0697.65013
  13. [13] S. Novotny, J. Ortiz and D.A. Narayan, Minimal k-rankings and the rank number of P²ₙ, Inform. Process. Lett. 109 (2009) 193-198, doi: 10.1016/j.ipl.2008.10.004. Zbl1286.05050
  14. [14] A. Sen, H. Deng and S. Guha, On a graph partition problem with application to VLSI layout, Inform. Process. Lett. 43 (1992) 87-94, doi: 10.1016/0020-0190(92)90017-P. Zbl0764.68132
  15. [15] X. Zhou, N. Nagai and T. Nishizeki, Generalized vertex-rankings of trees, Inform. Process. Lett. 56 (1995) 321-328, doi: 10.1016/0020-0190(95)00172-7. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.