Let F = {G1, …, Gt}
be a family of n-vertex graphs defined on the same vertex-set V, and
let k be a positive integer. A subset of vertices D ⊂ V is
called an (F, k)-core if, for each v ∈ V and for each
i = 1, …, t, there are at least k neighbours of v
in Gi that belong to
D. The subset D is called a connected (F, k)-core
if the subgraph induced by D in each Gi
is connected. Let δi be the minimum degree of Gi
and let δ(F) = minti=1δi. Clearly, an
(F, k)-core exists if and only if δ(F) [ges ] k,
and a connected (F, k)-core exists if and only if
δ(F) [ges ] k and each Gi is connected.
Let c(k, F) and cc(k, F)
be the minimum size of an (F, k)-core and a connected (F, k)-core,
respectively. The following asymptotic results are proved for every
t < ln ln δ and k < √lnδ:formula hereThe results are asymptotically tight for infinitely many families F. The results unify and
extend related results on dominating sets, strong dominating sets and connected dominating
sets.
Connected Dominating Sets in Wireless Networks with Different Transmission Ranges