## Abstract

Given a graph G and a pair (F1,F2) of graph families, the function GDISJG,F_{1},F_{2} takes as input, two induced subgraphs G_{1} and G_{2} of G, such that G_{1} ϵ F_{1} and G_{2} ϵ F_{2} and returns 1 if V (G_{1}) / V (G_{2}) = theta; 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 GDISJG,F_{1},F_{2} . A concept related to communication protocols is that of a (F_{1},F_{2})-separating family of a graph G. A collection F of subsets of V (G) is called a (F_{1},F_{2})-separating family for G, if for any two vertex disjoint induced subgraphs G_{1} ϵ F_{1},G_{2} ϵ F_{2}, there is a set F ϵ F with V (G_{1})F and V (G_{2}) / F = theta;. Given a graph G on n vertices, for any pair (F_{1},F_{2}) of hereditary graph families with sublinear communication complexity for GDISJG,F_{1},F_{2} , we give an enumeration algorithm that finds a subexponential sized (F_{1},F_{2})-separating family. In fact, we give an enumeration algorithm that finds a 2o(k)nO(1) sized (F_{1},F_{2})-separating family; where k denotes the size of a minimum sized set S of vertices such that V (G) \ S has a bipartition (V_{1}, V_{2}) with G[V_{1}] ϵ F_{1} and G[V_{2}] ϵ 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.

Original language | English |
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Title of host publication | 42nd International Symposium on Mathematical Foundations of Computer Science, MFCS 2017 |

Editors | Kim G. Larsen, Jean-Francois Raskin, Hans L. Bodlaender |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959770460 |

DOIs | |

State | Published - 1 Nov 2017 |

Externally published | Yes |

Event | 42nd International Symposium on Mathematical Foundations of Computer Science, MFCS 2017 - Aalborg, Denmark Duration: 21 Aug 2017 → 25 Aug 2017 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 83 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 42nd International Symposium on Mathematical Foundations of Computer Science, MFCS 2017 |
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Country/Territory | Denmark |

City | Aalborg |

Period | 21/08/17 → 25/08/17 |

## Keywords

- Communication Complexity
- FPT algorithms
- Separating Family

## ASJC Scopus subject areas

- Software