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Interval-Like Graphs and Digraphs

Authors Pavol Hell, Jing Huang, Ross M. McConnell, Arash Rafiey



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Author Details

Pavol Hell
  • School of Computing Science, Simon Fraser University, Burnaby, B.C., Canada V5A 1S6
Jing Huang
  • Mathematics and Statistics, University of Victoria, Victoria, B.C., Canada V8W 2Y2
Ross M. McConnell
  • Computer Science Department, Colorado State University, Fort Collins, CO 80523-1873
Arash Rafiey
  • Mathematics and Computer Science, Indiana State University, Terre Haute, IN 47809

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Pavol Hell, Jing Huang, Ross M. McConnell, and Arash Rafiey. Interval-Like Graphs and Digraphs. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 68:1-68:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.MFCS.2018.68

Abstract

We unify several seemingly different graph and digraph classes under one umbrella. These classes are all, broadly speaking, different generalizations of interval graphs, and include, in addition to interval graphs, adjusted interval digraphs, threshold graphs, complements of threshold tolerance graphs (known as `co-TT' graphs), bipartite interval containment graphs, bipartite co-circular arc graphs, and two-directional orthogonal ray graphs. (The last three classes coincide, but have been investigated in different contexts.) This common view is made possible by introducing reflexive relationships (loops) into the analysis. We also show that all the above classes are united by a common ordering characterization, the existence of a min ordering. We propose a common generalization of all these graph and digraph classes, namely signed-interval digraphs, and show that they are precisely the digraphs that are characterized by the existence of a min ordering. We also offer an alternative geometric characterization of these digraphs. For most of the above graph and digraph classes, we show that they are exactly those signed-interval digraphs that satisfy a suitable natural restriction on the digraph, like having a loop on every vertex, or having a symmetric edge-set, or being bipartite. For instance, co-TT graphs are precisely those signed-interval digraphs that have each edge symmetric. We also offer some discussion of future work on recognition algorithms and characterizations.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatoric problems
  • Mathematics of computing → Graph theory
Keywords
  • graph theory
  • interval graphs
  • interval bigraphs
  • min ordering
  • co-TT graph

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