Optimal accessing and non-accessing structures for graph protocols

An accessing set in a graph is a subset B of vertices such that ∃D ⊆ B, ∀v ∈ V \ B, |N (v) ∩ D| = 0 mod 2. In this paper, we introduce new bounds on the minimal size κ(G) of an accessing set, and on the maximal size κ(G) of a non-accessing set of a graph G. We show strong connections with perfect codes and give explicitly κ(G) and κ(G) for several families of graphs. Finally, we show that the corresponding decision problems are NP-Complete.