Discussiones Mathematicae Graph Theory 33(3) (2013)
531-557
DOI: https://doi.org/10.7151/dmgt.1707
Counting Maximal Distance-independent Sets in Grid Graphs
Reinhardt Euler
Université Européenne de Bretagne and Lab-STICC, CNRS, UMR 6285 | Paweł Oleksik
AGH University of Science and Technology |
Zdzisław Skupień
AGH Kraków |
Abstract
Previous work on counting maximal independent sets for paths and certain 2-dimensional grids is extended in two directions: 3-dimensional grid graphs are included and, for some/any l ∈ ℕ, maximal distance-l independent (or simply: maximal l-independent) sets are counted for some grids. The transfer matrix method has been adapted and successfully applied.
Keywords: independent set, grid graph, Fibonacci, Padovan numbers, transfer matrix method
2010 Mathematics Subject Classification: 11B39, 11B83, 05A15, 05C69.
References
[1] | R. Euler, Fibonacci number of a grid graph and a new class of integer sequences, J. Integer Seq. 8 (2005) Article 05.2.6. |
[2] | Z. Füredi, The number of maximal independent sets in connected graphs, J. Graph Theory 11 (1987) 463–470. |
[3] | Z. Skupień, Independence or domination. Positioning method in recursive counting on paths or cycles, a manuscript (2012). |
[4] | N.J.A. Sloane, The On-line Encyclopedia of Integer Sequences, (2007). www.research.att.com/˜njas/sequences/ |
[5] | R.P. Stanley, Enumerative Combinatorics (Cambridge Univ. Press, vol. 1, 1997), doi: 10.1017/CBO9780511805967 |
Received 31 January 2012
Revised 11 October 2012
Accepted 5 November 2012
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