## Abstract

Hierarchical graphs are used in order to describe systems with a sequential composition of sub-systems. A hierarchical graph consists of a vector of subgraphs. Vertices in a subgraph may “call” other subgraphs. The reuse of subgraphs, possibly in a nested way, causes hierarchical graphs to be exponentially more succinct than equivalent flat graphs. Early research on hierarchical graphs and the computational price of their succinctness suggests that there is no strong correlation between the complexity of problems when applied to flat graphs and their complexity in the hierarchical setting. That is, the complexity in the hierarchical setting is higher, but all “jumps” in complexity up to an exponential one are exhibited, including no jumps in some problems. We continue the study of the complexity of algorithms for hierarchical graphs, with the following contributions: (1) In many applications, the subgraphs have a small, often a constant, number of exit vertices, namely vertices from which control returns to the calling subgraph. We offer a parameterized analysis of the complexity and point to problems where the complexity becomes lower when the number of exit vertices is bounded by a constant. (2) We describe a general methodology for algorithms on hierarchical graphs. The methodology is based on an iterative compression of subgraphs in a way that maintains the solution to the problems and results in subgraphs whose size depends only on the number of exit vertices, and (3) We handle labeled hierarchical graphs, where edges are labeled by letters from some alphabet, and the problems refer to the languages of the graphs.

Original language | American English |
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Title of host publication | Descriptional Complexity of Formal Systems - 19th IFIP WG 1.02 International Conference, DCFS 2017, Proceedings |

Editors | Giovanni Pighizzini, Cezar Campeanu |

Publisher | Springer Verlag |

Pages | 114-127 |

Number of pages | 14 |

ISBN (Print) | 9783319602516 |

DOIs | |

State | Published - 2017 |

Event | 19th International Conference of Descriptional Complexity of Formal Systems, DCFS 2017 - Milano, Italy Duration: 3 Jul 2017 → 5 Jul 2017 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 10316 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 19th International Conference of Descriptional Complexity of Formal Systems, DCFS 2017 |
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Country/Territory | Italy |

City | Milano |

Period | 3/07/17 → 5/07/17 |

### Bibliographical note

Funding Information:The research leading to these results has received funding from the European Research Council under the European Union’s 7th Framework Programme (FP7/2007-2013, ERC grant no. 278410).

Publisher Copyright:

© IFIP International Federation for Information Processing 2017.