Independent protection in graphs

Mobile guards on the vertices in an independent set X of a graph G are used to defend G against an infinite sequence of attacks on up to k independent vertices at a time. If such a set X exists, then G is said to be k - i -protectable. The k -independent protection number i k ∞ ( G ) is the smallest number of guards required to protect G against an arbitrary sequence of such simultaneous attacks. sider the problem of determining classes of k - i -protectable graphs. We introduce a colouring scheme called multicolourings as aid to determine whether a graph is k - i -protectable, and to describe a protection strategy if it is. w that any well-covered graph (i.e., all maximal independent sets have the same cardinality k ) is k - i -protectable, even cycles in general are 1 - i -protectable but not 2 - i -protectable, and odd cycles in general are 2 - i -protectable but not 3 - i -protectable. We characterize k - i -protectable trees and give methods to construct larger k - i -protectable graphs from smaller ones.