Rank numbers of grid graphs

Abstract

a b s t r a c tA ranking of a graph is a labeling of the vertices with positive integers such that any path between vertices of the same label contains a vertex of greater label. The rank number of a graph is the smallest possible number of labels in a ranking. We find rank numbers of the Möbius ladder, K s × P n , and P 3 × P n . We also find bounds for rank numbers of general grid graphs P m × P n .

Rank numbers for some trees and unicyclic graphs

“…Note that the label for x is greater than or equal to all of the labels in H 1 and the ranking of H 2 is unchanged in the relabeling. If l > m, z is a vertex of H 2 labeled l and P (2) is an (x, z)-path in the graph G − H 1 , it has a vertex with label greater than l. If this does not hold we can consider an arbitrary (y, x) path P (1) in the graph G − H 2 ; the (y, z)-path consisting of P (1) followed by P (2) contradicts the definition of a ranking (in the original labeling of G). Hence it follows that our relabeling is a k-ranking of G.…”

“…However, rank numbers have been determined for several families of graphs including: paths, cycles, split graphs, complete multipartite graphs, Möbius ladder graphs, caterpillars, powers of paths and cycles, and some grid graphs [1][2][3][4][6][7][8]12], and [13]. A problem of interest is determining the rank number of a tree.…”

Maximizing the number of edges in optimal k-rankings

“…Bodlaender et al proved that given a bipartite graph G and a positive integer n, deciding whether a rank number of * A part of this work was performed at The Citadel and supported by The Citadel Foundation, and a second part of the work was performed at University of South Carolina Sumter, Corresponding address: Department of Mathematics and Computer Science, The Citadel, Charleston, SC 29409, Email: rigo.florez@citadel.edu G is less than n is NP-complete [2]. The rank number of paths, cycles, split graphs, complete multipartite graphs, powers of paths and cycles, and some grid graphs are well known [1,2,3,4,6,9,10].…”

Untitled

“…The determination of the rank number and the arank number was shown to be NP-complete [Laskar and Pillone 2000]. The rank number was explored in [Bodlaender et al 1998] where the authors showed that χ r (P n ) = log 2 n + 1. Rank numbers are known for a few other graph families such as cycles, wheels, complete bipartite graphs, and split graphs [Ghoshal et al 1996;Dereniowski 2004].…”

“…Leiserson 1980;Laskar and Pillone 2001;Sen et al 1992]. Numerous related papers have since followed [Bodlaender et al 1998;Hsieh 2002;Jamison 2003;Dereniowski 2006;Dereniowski and Nadolski 2006;Kostyuk and Narayan ≥ 2010;Kostyuk et al 2006;Isaak et al 2009;Novotny et al 2009a]. Ghoshal, Laskar, and Pillone were the first to investigate minimal k-rankings [Ghoshal et al 1999;1996;Laskar and Pillone 2001;.…”