on •-lict signed graphs l_•_c(s) and •-line signed graphs l_•(s)

A signed graph (or‎, ‎in short‎, sigraph) $s=(s^u,sigma)$ consists of an underlying graph $s^u‎ :‎=g=(v,e)$ and a function $sigma:e(s^u)longrightarrow {+,}$‎, ‎called the signature of $s$‎. ‎a marking of $s$ is a function $mu:v(s)longrightarrow {+,}$‎. ‎the canonical marking of a signed graph $s$‎, ‎denoted $mu_sigma$‎, ‎is given as $$mu_sigma(v)‎ :‎= prod_{vwin e(s)}sigma(vw).$$‎ ‎the line graph of a graph $g$‎, ‎denoted $l(g)$‎, ‎is the graph in which edges of $g$ are represented as vertices‎, ‎two of these vertices are adjacent if the corresponding edges are adjacent in $g$‎. ‎there are three notions of a line signed graph of a signed graph $s=(s^u,sigma)$ in the literature‎, ‎viz.‎, ‎$l(s)$‎, ‎$l_times(s)$ and $l_bullet(s)$‎, ‎all of which have $l(s^u)$ as their underlying graph only the rule to assign signs to the edges of $l(s^u)$ differ‎. ‎every edge $ee' $ in $l(s)$ is negative whenever both the adjacent edges $e$ and $e' $ in s are negative‎, ‎an edge $ee' $ in $l_times(s)$ has the product $sigma(e)sigma(e' )$ as its sign and an edge $ee' $ in $l_bullet(s)$ has $mu_sigma(v)$ as its sign‎, ‎where $vin v(s)$ is a common vertex of edges $e$ and $e' $‎. ‎‎the linecut graph (or‎, ‎in short‎, lict graph) of a graph $g=(v,e)$‎, ‎denoted by $l_c(g)$‎, ‎is the graph with vertex set $e(g)cup c(g)$‎, ‎where $c(g)$ is the set of cutvertices of $g$‎, ‎in which two vertices are adjacent if and only if they correspond to adjacent edges of $g$ or one vertex corresponds to an edge $e$ of $g$ and the other vertex corresponds to a cutvertex $c$ of $g$ such that $e$ is incident with $c$‎. ‎‎in this paper‎, ‎we introduce dotlict signed graph (or $bullet$lict signed graph} $l_{bullet_c}(s)$‎, ‎which has $l_c(s^u)$ as its underlying graph‎. ‎every edge $uv$ in $l_{bullet_c}(s)$ has the sign $mu_sigma(p)$‎, ‎if $u‎, ‎v in e(s)$ and $pin v(s)$ is a common vertex of these edges‎, ‎and it has the sign $mu_sigma(v)$‎, ‎if $uin e(s)$ and $vin c(s)$‎. ‎we characterize signed graphs on $k_p$‎, ‎$pgeq2$‎, ‎on cycle $c_n$ and on $k_{m,n}$ which are $bullet$lict signed graphs or $bullet$line signed graphs‎, ‎characterize signed graphs $s$ so that $l_{bullet_c}(s)$ and $l_bullet(s)$ are balanced‎. ‎we also establish the characterization of signed graphs $s$ for which $ssim l_{bullet_c}(s)$‎, ‎$ssim l_bullet(s)$‎, ‎$eta(s)sim l_{bullet_c}(s)$ and $eta(s)sim l_bullet(s)$‎, ‎here $eta(s)$ is negation of $s$ and $sim$ stands for switching equivalence‎.