Communication Complexity of Pairs of Graph Families with Applications

Given a graph G and a pair (\mathcal{F}_1,\mathcal{F}_2) of graph families, the function {\sf GDISJ}_{G,{\cal F}_1,{\cal F}_2} takes as input, two induced subgraphs G_1 and G_2 of G, such that G_1 \in \mathcal{F}_1 and G_2 \in \mathcal{F}_2 and returns 1 if V(G_1)\cap V(G_2)=\emptyset 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,\cal F_1,\cal F_2. A concept related to communication protocols is that of a (\mathcal{F}_1,\mathcal{F}_2)-separating family of a graph G. A collection \mathcal{F} of subsets of V(G) is called a (\mathcal{F}_1,\mathcal{F}_2)-separating family} for G, if for any two vertex disjoint induced subgraphs G_1\in \mathcal{F}_1,G_2\in \mathcal{F}_2, there is a set F \in \mathcal{F} with V(G_1) \subseteq F and V(G_2) \cap F = \emptyset. Given a graph G on n vertices, for any pair (\mathcal{F}_1,\mathcal{F}_2) of hereditary graph families with sublinear communication complexity for GDISJ_G,\cal F_1,\cal F_2, we give an enumeration algorithm that finds a subexponential sized (\mathcal{F}_1,\mathcal{F}_2)-separating family. In fact, we give an enumeration algorithm that finds a 2^{o(k)}n^{\Oh(1)} sized (\mathcal{F}_1,\mathcal{F}_2)-separating family; where k denotes the size of a minimum sized set S of vertices such that V(G)\setminus S has a bipartition (V_1,V_2) with G[V_1] \in {\cal F}_1 and G[V_2]\in {\cal F}_2. 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.