C-consistent and C-cycle compatible dot-line signed graphs

A signed graphS=(Su,σ) has an underlying graph Su:=G=(V,E) and a function σ:E(Su)→{+,−}. A marking of S is a function μ:V(S)→{+,−}. In canonical marking, denoted μσ, we assign +('−') sign to a vertex if its negative degree is even(odd). The dot-line (or
- -line) signed graph of S, denoted L
- (S), is obtained by representing edges of S as vertices, two of these vertices are adjacent if the corresponding edges are adjacent in S and edge ee′ in L
- (S) is negative whenever negative degree of a common vertex of edges e and e′ in S is odd. S is called C-consistent if every cycle in S has an even number of negative vertices under canonical marking. S is called C-cycle compatible if for every cycle Z in S, the product of signs of its vertices equals the product of signs of its edges with respect to canonical marking. In this paper, we establish structural characterizations of signed graph S so that L
- (S) is C-consistent and C-cycle compatible.