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Simultaneous Representation of Proper and Unit Interval Graphs

Authors Ignaz Rutter , Darren Strash , Peter Stumpf , Michael Vollmer

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Author Details

Ignaz Rutter
  • Faculty of Computer Science and Mathematics, University of Passau, Germany
Darren Strash
  • Department of Computer Science, Hamilton College, Clinton, NY, USA
Peter Stumpf
  • Faculty of Computer Science and Mathematics, University of Passau, Germany
Michael Vollmer
  • Department of Informatics, Karlsruhe Institute of Technology (KIT), Germany

Funding

This work was partially supported by grant RU 1903/3-1 of the German Research Foundation (DFG) and by the DFG Research Training Group 2153: "Energy Status Data - Informatics Methods for its Collection, Analysis and Exploitation".

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Ignaz Rutter, Darren Strash, Peter Stumpf, and Michael Vollmer. Simultaneous Representation of Proper and Unit Interval Graphs. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 80:1-80:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.ESA.2019.80

Abstract

In a confluence of combinatorics and geometry, simultaneous representations provide a way to realize combinatorial objects that share common structure. A standard case in the study of simultaneous representations is the sunflower case where all objects share the same common structure. While the recognition problem for general simultaneous interval graphs - the simultaneous version of arguably one of the most well-studied graph classes - is NP-complete, the complexity of the sunflower case for three or more simultaneous interval graphs is currently open. In this work we settle this question for proper interval graphs. We give an algorithm to recognize simultaneous proper interval graphs in linear time in the sunflower case where we allow any number of simultaneous graphs. Simultaneous unit interval graphs are much more "rigid" and therefore have less freedom in their representation. We show they can be recognized in time O(|V|*|E|) for any number of simultaneous graphs in the sunflower case where G=(V,E) is the union of the simultaneous graphs. We further show that both recognition problems are in general NP-complete if the number of simultaneous graphs is not fixed. The restriction to the sunflower case is in this sense necessary.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Intersection Graphs
  • Recognition Algorithm
  • Proper/Unit Interval Graphs
  • Simultaneous Representations

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