Characterization of Vertex labeling of 1-uniform dcsl graph which form a lattice

A 1-uniform dcsl of a graph G is an injective set-valued function f : V (G) → 2X , X be a non-empty ground set, such that the corresponding induced function f⊕ : V (G)×V (G) → 2X \ {φ} satisfies | f⊕(u,v)| = 1.d(u,v) for all distinct u,v ∈ V (G), where d(u,v) is the distance between u and v. Let F 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 F is a sequence P0 = P,P1,P2 . . .Pn = Q in F such that d(P,Q) =| P△Q |= n and d(Pi,Pi+1) = 1 for 0 ≤ i ≤ n−1. The family F is well-graded (or wg-family), if there is a tight path between any two of its distinct sets. Any family F of subsets of X defines a graph GF = (F ,EF ), where EF = {{P,Q} ⊆ F :| P△Q |= 1}, and we call GF , an F -induced graph. In this paper, we study 1-uniform dcsl graphs whose vertex labelings whether or not forms a lattice and prove that the cover graph CF of a poset F with respect to set inclusion ‘⊆’ is isomorphic to the F -induced graph GF .


Introduction
Acharya [1] introduced the notion of vertex set-valuation as a set-analogue of number valuation.For a graph G = (V, E) and a non empty set X, Acharya defined a set-valuation of G as an injective set-valued function f : V (G) → 2 X , and he defined a set-indexer f ⊕ : E(G) → 2 X \ {ϕ }as a set-valuation such that the function given by f ⊕ (uv) = f (u) ⊕ f (v) for every uv ∈ E(G) is also injective, where 2 X is the set of all the subsets of X and '⊕' is the binary operation of taking the symmetric difference of subsets of X. Acharya and Germina [2], who has been studying topological set-valuation, introduced the particular kind of setvaluation for which a metric, especially the cardinality of the symmetric difference, is associated with each pair of vertices in proportion to the distance between them in the graph [2].In otherwords, the question is whether one can determine those graphs G = (V, E) that admit an injective set-valued function f : V (G) → 2 X , X being a non empty ground set such that the cardinality of the symmetric difference f ⊕ (uv) is proportional to the usual path distance d G (u, v) between u and v in G, for each pair of distinct vertices u and v in G.They [2] called such a set-valuation f of G a distance-compatible set-labeling (dcsl) of G, and the ordered pair (G, f ), a distance-compatible set-labeled (dcsl) graph.The dcsl graph (G, f ), whose corresponding ground set is called a dcsl-set.The following universal theorem has been established .Theorem 1.1.[2] Every graph admits a dcsl.
A dcsl f of G is k-uniform if all the constant 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.A family of sets F is well-graded if any two sets in F can be connected by a sequence of sets formed by single element insertion and deletion, without redundant operations, such that all intermediate sets in the sequence belong to F .Well-graded families are of interest for theorist in several different areas of combinatorics, as various families of sets or relations are well-graded.Using representation theorems, well-graded families are applied to the partial cubes [10,9,11,12], and to the oriented media which are semigroups of trasformations satisfying certain axioms (see [13,14]).Definition 1.1.[4] Let F 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 F is a sequence P 0 = P, P 1 , P 2 . . .
The family F is well-graded (or wg-family), if there is a tight path between any two of its distinct sets.Any family F of subsets of X defines a graph G F = (F , E F ), where E F = {{P, Q} ⊆ F :| P △ Q |= 1}, and we call G F , an F -induced graph.
One may recall a partially ordered set (or a poset, in short) as a structure (P, ≼) where P is a non-empty set and "≼" is a partial order relation on P such that '≼' is reflexive, antisymmetric and transitive.We denote (x, y) ∈ P by x ≼ y.We say that z covers y if and only if y ≺ z and y ≼ x ≼ z implies either x = y or x = z.A Hasse diagram of a poset (P, ≼) is a drawing in which the points of P are placed so that if y covers x, then y is placed at a higher level than x and joined to x by a line segment.The corresponding graph is called the Hasse graph of the poset.A Cover graph [8] of a poset (P, ≼) is the graph with vertex set P such that x, y ∈ P are adjacent if and only if one of them covers the other.A poset (L, ≼) is a lattice if every pair of elements x, y ∈ L, has a least upper bound (lub, for short), denoted by x ∨ y (called join), and a greatest lower bound (glb, for short), denoted by x ∧ y (called meet).In general, a lattice is denoted by (L, ≼).Throughout this paper lattice (and poset) means lattice (and poset) under set inclusion ⊆.Unless otherwise mentioned, for all the terminology in graph theory and lattice theory, the reader is referred, respectively to [6,5].This paper initiates a study on how the structure of a 1-uniform dcsl graph is related to a lattice.The set of vertex labelings of a 1-uniform dcsl path forms a lattice.However, in general the set of vertex labelings of all 1-uniform dcsl graph do not form a lattice.
Proof.Let F be a well-graded family of subsets of X.Clearly, F is a poset.Define I : Remark 2.2.By the Theorem 2.4, all F -induced graphs G F are 1-uniform dcsl graphs.Also by Proposition 2.2, C F ∼ = G F .Hence, the cover graph C F is a 1-uniform dcsl graph.

Conclusion
We have established that the cover graph C F of a wg-family F (which is a poset with respect to set inclusion '⊆') is isomorphic to F -induced graph G F , so that one can easily calculate the dimension [15] of a F -induced graph G F .Also by invoking Theorem 2.4, one may note that the F -induced graph G F always admit 1-uniform dcsl, and hence can compute the dimension of 1-uniform dcsl.
Following is an open problem for further investigation.
Problem 1: Characterize the structural properties of a 1-uniform dcsl graphs whose set of vertex labeling form a lattice.Also, for given lattice characterize those 1-uniform dcsl labeled graph with the collection of sets of lattice, as vertex labeling.
and only if either I(A) covers I(B) or I(B) covers I(A).Suppose | A △ B |= 1.With out loss of generality, let A ⊆ B. Then |A| = |B| − 1.Thus, I(A) ⊆ I(B) and |I(A)| = |I(B)| − 1.In this case I(B) covers I(A) in the poset F , for, if possible suppose I(B) do not covers I(A), then there exists I(C) ∈ F such that I(A) ⊂ I(C) ⊂ I(B) and |I(A)| = |I(B)| − n(n ≥ 2), a contradiction.Conversely, suppose I(B) covers I(A) in the poset F ( Similar argument hold if I(A) covers I(B) ).Claim: |A|