Total minimal dominating signed graph

A Smarandachely k-signed graph (Smarandachely k-marked graph) is an ordered pair S = (G, [sigma]) (S = (G, [mu])) where G = (V, E) is a graph called underlying graph of S and [sigma] : E [right arrow] ([[bar.e].sub.1], [[bar.e].sub.2], ..., [[bar.e].sub.k]) ([mu] : V [right arrow] ([[bar.e].sub.1], [[bar.e].sub.i], ..., [[bar.e].sub.k])) is a function, where each [[bar.e].sub.i] [member of] {+, -}. Particularly, a Smarandachely 2-signed graph or Smarandachely 2-marked graph is called abbreviated a signed graph or a marked graph. In this paper, we define the total minimal dominating signed graph [M.sub.t] (S) = ([M.sub.t](G), [sigma]) of a given signed digraph S = (G, [sigma]) and offer a structural characterization of total minimal dominating signed graphs. Further, we characterize signed graphs S for which S ~ [M.sub.t] (S) and L(S) ~ [M.sub.t] (S), where ~ denotes switching equivalence and [M.sub.t] (S) and L(S) are denotes total minimal dominating signed graph and line signed graph of S respectively. Key Words: Smarandachely k-signed graphs, Smarandachely k-marked graphs, signed graphs, marked graphs, balance, switching, total minimal dominating signed graph, line signed graphs, negation. AMS(2000): 05C22