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

Abstract

A 1-uniform dcsl of a graph $G$ is an injective set-valued function $f : V(G)\rightarrow 2^{X}, X$ be a non-empty ground set, such that the corresponding induced function $f^{\oplus} :V(G) \times V(G) \rightarrow 2^{X}\setminus \{\phi\}$ satisfies $\vert f^{\oplus}(u, v)\vert = 1.d(u, v)$ for all distinct $u, v \in V(G)$, where $d(u, v)$ is the distance between $u$ and $v$. Let ${\mathscr{F}}$ be a family of subsets of a set $X$. A {\it tight path} between two distinct sets $P$ and $Q$ (or from $P$ to $Q$) in ${\mathscr{F}}$ is a sequence $P_0 = P, P_1, P_2 \dots P_n = Q$ in ${\mathscr{F}}$ such that $d(P,Q)= \mid P \bigtriangleup Q \mid = n$ and $d(P_i, P_{i+1}) = 1$ for $0 \leq i\leq n-1$. The family ${\mathscr{F}}$ is {\it well-graded} (or {\it wg-family}), if there is a {\it tight path} between any two of its distinct sets. Any family ${\mathscr{F}}$ of subsets of $X$ defines a graph $G_{\mathscr{F}} = ( {\mathscr{F}} , E_{\mathscr{F}})$, where $E_{\mathscr{F}} = \{ \{P,Q \} \subseteq {\mathscr{F}}:\mid P \bigtriangleup Q \mid = 1 \} $, and we call $G_{\mathscr{F}$, an ${\mathscr{F}}$-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 $C_{{\mathscr{F}}}$ of a poset ${\mathscr{F}}$ with respect to set inclusion ` $\subseteq$' is isomorphic to the ${\mathscr{F}}$-induced graph $G_{{\mathscr{F}}}$.