Communication complexity and graph families

Given a graph G and a pair (F₁, F₂) of graph families, the function GDISJ_{G, F1, F2} takes as input, two induced subgraphsG₁ andG₂ ofG, such thatG₁ ∈ F₁ andG₂ ∈ F₂ and returns 1 ifV(G₁) ∩ V(G₂) = ∅ and 0 otherwise. We study the communication complexity of this problem in the two-party model. In particular, we look at pairs of hereditary graph families. We show that the communication complexity of this function, when the two graph families are hereditary, is sublinear if and only if there are finitely many graphs in the intersection of these two families. Then, using concepts from parameterized complexity, we obtain nuanced upper bounds on the communication complexity of GDISJ_{G, F1, F2}. A concept related to communication protocols is that of a (F₁, F₂)-separating family of a graph G. A collection F of subsets of V(G) is called a (F₁, F₂)-separating family for G, if for any two vertex disjoint induced subgraphs G₁ ∈ F₁,G₂ ∈ F₂, there is a set F ∈ F with V(G₁) ⊆ F andV(G₂) ∩ F = ∅. Given a graphG onn vertices, for any pair (F₁, F₂) of hereditary graph families with sublinear communication complexity for GDISJ_{G, F1, F2}, we give an enumeration algorithm that finds a subexponential sized (F₁, F₂)-separating family. In fact, we give an enumeration algorithm that finds a 2^o(k⁾n^O(1⁾ sized (F₁, F₂)-separating family, where k denotes the size of a minimum sized set S of vertices such that V(G) \ S has a bipartition (V₁,V₂) with G[V₁] ∈ F₁ and G[V₂] ∈ F₂. We exhibit a wide range of applications for these separating families, to obtain combinatorial bounds, enumeration algorithms, as well as exact and FPT algorithms for several problems.