A computational study of f-reversible processes on graphs

Abstract An f -reversible process on a graph  G = ( V , E ) is a graph dynamical system on  V ( G ) defined as follows. Given a function  f : V ( G ) → N and an initial vertex labeling  c 0 : V ( G ) → { 0 , 1 } , every vertex  v changes its label if and only if at least  f ( v ) of its neighbors have the opposite state, synchronously in discrete-time. For such processes, we present a new nondecreasing time function similar to the monotonically decreasing energy functions used to study threshold networks, which leads to a periodic behavior after a transient phase. Using this new function, we provide a tight upper bound on the transient length of  f -reversible processes. Furthermore, we prove that it is equal to  n − 3 for trees with  n ≥ 4 vertices and  Im ( f ) = { 2 } . Moreover we present an algorithm that generates all the initial configurations attaining this bound and we prove that the number of such configurations is  O ( n ) . We also consider the problem of determining the smallest number  r f ( G ) of vertices with initial label 1 for which all the vertices eventually reach label 1 after the complete evolution of the dynamics, which models consensus problems on networks. We prove that it is N P -hard to compute  r f ( G ) even for bipartite graphs with  Δ ( G ) ≤ 3 and  Im ( f ) = { 1 , 2 , 3 } . Finally, we prove that  β ( G ) ≤ r f ( G ) ≤ β ( G ) + 1 , when  f ( v ) = d ( v ) for all  v ∈ V ( G ) , where  β ( G ) is the size of a minimum vertex cover of  G .