Characterization of Vertex labeling of 1-uniform dcsl graph which form a lattice
K. A. Germina, K. Nageswararao · Journal of Fuzzy Set Valued Analysis · 2015
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 $G_{{\\mathscr{F}}}$.