Abstract:

The graphs G=(V,E) considered here are finite, simple and without isolated vertices. For a function f:V∪E→{－1,1},weight of 〖WTBX〗f〖WTBZ〗 is ω(f)=∑〖DD(X〗x∈V∪E〖DD)〗 f(x). For each element 〖WTBX〗x〖WTBZ〗 in V∪E, we define f［x］=∑〖DD(X〗y∈〖WTBX〗N〖WTBX〗T(x)〖DD)〗f(y), where 〖WTBX〗NT〖WTBZ〗(〖WTBX〗x〖WTBZ〗) denotes adjacent and incident element of 〖WTBX〗x〖WTBZ〗. A total signed local dominating function (TSLDF) of 〖WTBX〗G〖WTBZ〗 is a function f:V∪E→{－1,1} so that f［x］≥1 for all x∈V∪E. The total signed local domination number of 〖WTBX〗G〖WTBZ〗 is the minimum weight of a TSLDF on 〖WTBX〗G〖WTBZ〗, denoted by γTsl(G). In this paper, some lower bounds in general graphs and an upper bound in complete bipartite graph 〖WTBX〗Km,n〖WTBZ〗 for γTsl, and the exact value on γTsl of cycle 〖WTBX〗Cn〖WTBZ〗 can be obtained.

Key words:  complete bipartite graph; cycle; total signed local domination number; lower bound; upper bound