Advances and Applications in Discrete Mathematics
Volume 15, Issue 2, Pages 113 - 123
(April 2015) http://dx.doi.org/10.17654/AADMApr2015_113_123 |
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CHARACTERIZATION OF 1-UNIFORM dcsl GRAPHS AND WELL GRADED FAMILY OF SETS
K. A. Germina and Jinto James
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Abstract: A distance compatible set labeling (dcsl) of a connected graph G is an injective set assignment X being a non-empty ground set such that the corresponding induced function given by satisfies for every pair of distinct vertices where denotes the path distance between u and v and is a constant, not necessarily an integer, depending on the pair of vertices u, v chosen. A dcsl f of G is k-uniform if all the constants of proportionality with respect to f are equal to k, and if G admits such a dcsl, then G is called a k-uniform dcsl graph. Let be a family of subsets of a set X. A tight path between two distinct sets P and Q (or from P to Q) in is a sequence in such that and for The family is well-graded family (or wg-family), if there is a tight path between any two of its distinct sets. In this paper, we establish the relationship between 1-uniform dcsl graphs and wg-family of sets. |
Keywords and phrases: dcsl graphs, 1-uniform dcsl graphs, wg-family of sets. |
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