# Signed edge k-domination numbers in graphs

The closed neighborhood *N*_{G}[*e*] of an edge *e *in a graph *G *is the set consisting of *e *and of all edges having an end-vertex in common with *e*. Let *f *be a function on *E*(*G*), the edge set of *G*, into the set *{-1,1} *and let *k*≥*1** *be an integer. If Σ_{x∈}_{N}_{[e]}*f*(*x*) *≥ **k *for each edge *e **∈ **E*(*G*), then *f *is called a signed edge *k*-domination function (SE*k*DF) of *G*. The signed edge *k*-domination number γ’_{sk}(*G*) of *G *is defined as

γ’_{sk}(*G*) = min*{*Σ_{e∈E}_{(G)}*f*(*e*) | *f *is an SE*k*DF of G*}*

In this presentation, we calculate the signed edge *k*-domination numbers for complete graphs and complete bipartite graphs. We then show that, for any simple graph *G*,

γ’_{sk}(*G*) *≥ |**V *(*G*)*| **- **| **E*(*G*)*| *+ *k **- *1,

and characterize all graphs that achieve this lower bound. This generalizes the existing results for *k *= 1.