The complexity of open k-monopolies in graphs for negative k

The complexity of open k-monopolies in graphs for negative k

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Let (G) be a graph with vertex Feminine Wash set (V(G)), (delta(G)) minimum degree of (G) and (kinleft{1-leftlceilrac{delta(G)}{2}
ight
ceil,ldots ,leftlfloor rac{delta(G)}{2}
ight
floor
ight}).Given a nonempty set (Msubseteq V(G)) a vertex (v) of (G) is said to be (k)-controlled by (M) if (delta_M(v)gerac{delta_{V(G)}(v)}{2}+k) where (delta_M(v)) represents the number of neighbors of (v) in (M).The set (M) is called an open (k)-monopoly for (G) if it (k)-controls every vertex (v) of (G).In S2 H2OMELON this short note we prove that the problem of computing the minimum cardinality of an open (k)-monopoly in a graph for a negative integer (k) is NP-complete even restricted to chordal graphs.