On.-lict signed graphs L ∙ c (S) L∙c(S) and ∙-line signed graphs L ∙ (S) L∙(S)

A emph{signed graph} (or, in short, emph{sigraph}) S=(S u, sigma) S=(Su,sigma) consists of an underlying graph S u: =G=(V,E) Su:=G=(V,E) and a function sigma:E(S u)longrightarrow+,− sigma:E(Su)longrightarrow+,−, called the signature of S S. A emph{marking} of S S is a function mu:V(S)longrightarrow+,− mu:V(S)longrightarrow+,−. The emph{canonical marking} of a signed graph S S, denoted mu s igma musigma, is given as mu s igma(v):=prod vwinE(S) sigma(vw). musigma(v):=prodvwinE(S)sigma(vw). The line-cut graph (or, in short, emph{lict graph}) of a graph G=(V,E) G=(V,E), denoted by L c (G) Lc(G), is the graph with vertex set E(G)cupC(G) E(G)cupC(G), where C(G) C(G) is the set of cut-vertices of G G, in which two vertices are adjacent if and only if they correspond to adjacent edges of G G or one vertex corresponds to an edge e e of G G and the other vertex corresponds to a cut-vertex c c of G G such that e e is incident with c c. In this paper, we introduce emph{Dot-lict signed graph} (or emph{bullet bullet -lict signed graph}) L bullet c (S) Lbulletc(S), which has L c (S u) Lc(Su) as its underlying graph. Every edge uv uv in L bullet c (S) Lbulletc(S) has the sign mu s igma(p) musigma(p), if u,vinE(S) u,vinE(S) and pinV(S) pinV(S) is a common vertex of these edges, and it has the sign mu s igma(v) musigma(v), if uinE(S) uinE(S) and vinC(S) vinC(S). we characterize signed graphs on K p Kp, pgeq2 pgeq2, on cycle C n Cn and on K m,n Km,n which are bullet bullet -lict signed graphs or bullet bullet -line signed graphs, characterize signed graphs S S so that L bullet c (S) Lbulletc(S) and L b ullet(S) Lbullet(S) are balanced. We also establish the characterization of signed graphs S S for which SsimL bullet c (S) SsimLbulletc(S), SsimL b ullet(S) SsimLbullet(S), eta(S)simL bullet c (S) eta(S)simLbulletc(S) and eta(S)simL b ullet(S) eta(S)simLbullet(S), here eta(S) eta(S) is negation of S S and sim sim stands for switching equivalence.