on label graphoidal covering number-i

let $g=(v‎, ‎e)$ be a graph with $p$ vertices and $q$ edges‎. ‎an emph{acyclic‎ ‎graphoidal cover} of $g$ is a collection $psi$ of paths in $g$‎ ‎which are internally-disjoint and cover each edge of the graph‎ ‎exactly once‎. ‎let $f‎: ‎vrightarrow {1‎, ‎2‎, ‎ldots‎, ‎p}$ be a bijective‎ ‎labeling of the vertices of $g$‎. ‎let $uparrow!g_f$ be the‎ ‎directed graph obtained by orienting the edges $uv$ of $g$ from‎ ‎$u$ to $v$ provided $f(u)< f(v)$‎. ‎if the set $psi_f$ of all‎ ‎maximal directed paths in $uparrow!g_f$‎, ‎with directions‎ ‎ignored‎, ‎is an acyclic graphoidal cover of $g$‎, ‎then $f$ is called‎ ‎a emph{graphoidal labeling} of $g$ and $g$ is called a‎ ‎emph{label graphoidal graph} and $eta_l=min{|psi_f|‎: ‎f {rm‎ ‎is a graphoidal labeling of} g}$ is called the emph{label‎ ‎graphoidal covering number} of $g$‎. ‎in this paper we characterize‎ ‎graphs for which (i) $eta_l=q-m$‎, ‎where $m$ is the number of‎ ‎vertices of degree 2 and (ii) $eta_l= q$‎. ‎also‎, ‎we determine the value of label graphoidal covering number for unicyclic graphs‎.