Rank numbers for some trees and unicyclic graphs

Abstract

A ranking on a graph is an assignment of positive integers to its vertices such that any path between two vertices of the same rank contains a vertex of strictly larger rank. The rank number of a graph is the fewest number of labels that can be used in a ranking. In this paper we determine rank numbers for some trees and unicyclic graphs. (2000). Primary 05C78; Secondary 05C15. Mathematics Subject Classification

Untitled

“…Lemma 2 [Sergel et al 2011]. Let H 1 and H 2 be two vertex-disjoint graphs such that χ r (H 1 ) = χ r (H 2 ) = k. Let G be a connected supergraph of H 1 ∪ H 2 .…”

Graphs with large rank numbers and rank numbers of subdivided stars

“…The mathematical studies of vertex rankings were initiated by Ghoshal and Laskar in [5]. Since then, the rank number of numerous families of graphs have been established, for example see [6][7][8][9][10][11][12][13][14]. A generalization of k-ranking using the l p norm is discussed in [15].…”

Rank numbers of graphs that are combinations of paths and cycles

“…Interest in rankings of graphs was sparked by its many applications to other fields, including designs of very large scale integration (VLSI) layouts, Cholesky factorizations of matrices in parallel, and scheduling problems of assembly steps in manufacturing systems [Duff and Reid 1983;Iyer et al 1991;Leiserson 1980;Liu 1990;Sen et al 1992]. The optimal tree node ranking problem is identical to the problem of generating a minimum-height node separator tree for a tree.…”

“…Bodlaender et al [1995] show that for a graph G and a positive integer t, the question of whether χ r (G) ≤ t is NP-complete. However, the rank number of numerous families of graphs have been established [Alpert 2010;Bruoth and Horňák 1999;Dereniowski and Nadolski 2006;Hsieh 2002;Novotny et al 2009;Ortiz et al 2010;Sergel et al 2011]. Bodlaender et al [1995] established that χ r (P n ) = log 2 n + 1, where P n is a path on n vertices.…”

“…Let H be a subgraph of G. Then χ r (H ) ≤ χ r (G). Sergel et al 2011]. Let H 1 and H 2 be two vertex-disjoint graphs such that…”