On binary switching finite networks

We call a finite graph G = (V,E) a binary network if the state set of its nodes has only two elements, say, 0 and 1, representing respectively ‘OFF’ and ‘ON’ state. A switch at node v switches both the state of v and the state of each of its neighbors. It is shown in [1] that given any initial state of a network of order n > 3, we can always reach at a consistent status, i.e., either all the nodes are ON or all are OFF. In this paper we consider a more general problem: Given a subset S ⊂ V, can we reach to a state such that the state of each node within S is 1(or 0) while the states of nodes outside S is another? We present some sufficient conditions for some specific S that satisfies this condition.