Irreversible conversion processes with deadlines

Abstract Given a graph G, a deadline t d ( u ) and a time-dependent threshold f ( u , t ) for every vertex u of G, we study sequences C = ( c 0 , c 1 , … ) of 0/1-labelings c i of the vertices of G such that for every t ∈ N , we have c t ( u ) = 1 if and only if either c t − 1 ( u ) = 1 or at least f ( u , t − 1 ) neighbors v of u satisfy c t − 1 ( v ) = 1 . The sequence C models the spreading of a property/commodity within a network and it is said to converge to 1 on time, if c t d ( u ) ( u ) = 1 for every vertex u of G, that is, if every vertex u has the spreading property/received the spreading good by time t d ( u ) . We study the smallest number irr ( G , t d , f ) of vertices u with initial label c 0 ( u ) equal to 1 that result in a sequence C converging to 1 on time. If G is a forest or a clique, we present efficient algorithms computing irr ( G , t d , f ) . Furthermore, we prove lower and upper bounds relying on counting and probabilistic arguments. For special choices of t d and f, the parameter irr ( G , t d , f ) coincides with well-known graph parameters related to domination and independence in graphs.