Advances and Applications in Discrete Mathematics
Volume 16, Issue 2, Pages 89 - 109
(October 2015) http://dx.doi.org/10.17654/AADMOct2015_089_109 |
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DIMENSION OF VERTEX LABELING OF k-UNIFORM DCSL PATH
K. Nageswararao and K. A. Germina
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Abstract: Let an injective function where is the vertex set of a graph G and is the power set of a non-empty set X, be given. Consider the induced function defined by where denotes the symmetric difference of the two sets. The function f is called a k-uniform dcsl (and X a k-uniform dcsl-set) of the graph G, if there exists a constant k such that where is the length of a shortest path between u and v in G. If a graph G admits a k-uniform dcsl, then G is called a k-uniform dcsl graph. The k-uniform dcsl index of a graph G, denoted by is the minimum of the cardinalities of X, as X varies over all k-uniform dcsl-sets of G. A linear extension of a partial order is a linear order on the elements of such that in implies in for all A set of linear extensions of is a realizer of if, for every incomparable pair there are with in and in The dimension of denoted by is the minimum of the cardinalities of realizers of Let be the range of a k-uniform dcsl of the path on vertices. The purpose of this paper is to prove that whether or not forms a lattice with respect to set inclusion
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Keywords and phrases: 1-uniform distance-compatible set-labeling, k-uniform distance-compatible set-labeling, dcsl index, k-uniform dcsl index, dimension of the poset, partition of the poset, lattice. |
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