Minimal separating sets of maximum size

Minimal Separating Sets for

For a Muller automaton only a subset of its states is needed to decide whether a run is accepting or not: The set I the innnitely often visited states can be replaced by the intersection I \ W with a xed set W of states, provided W is large enough to distinguish between accepting and non-accepting loops in the automaton. We call such a subset W a separating set. Whereas the idea was previously ...

Minimal completely separating systems of sets

Let [n] denote an n-set. A subset,) of [n] i from j if i and j t/:. 5'. A collection of k-sets n called (n, k) completely if, for each ordered pair j) E [n] x [71,] with i n which i from j. Let denote the size of a smallest (71" k) completely Amongst other things, it will be shown that R( 71" k) for n > except when n (Hl) 1, and R(71" k) k + 1 for G) n k 2 /2. These results build on and extend ...

Minimal dominating sets in maximum domatic partitions

The domatic number d(G) of a graph G = (V,E) is the maximum order of a partition of V into dominating sets. Such a partition Π = {D1, D2, . . . , Dd} is called a minimal dominating d-partition if Π contains the maximum number of minimal dominating sets, where the maximum is taken over all d-partitions of G. The minimal dominating d-partition number Λ(G) is the number of minimal dominating sets ...

Finding all minimum-size separating vertex sets in a graph

We present a new algorithm based upon network flows for finding all minimum size separating vertex sets in an undirected graph. The sequential implementation of our algorithm runs in ®(mm(max(Mnk,knmmin(k,'In)),mQx(Mn,k2n3))) = 0 (2 kn3) time, where M is the number of minimum size separating vertex sets The parallel implementai 6 tion runs either in 0 (&logn) deterministic time using ©(M2n2 + k...

On maximum number of minimal dominating sets in graphs