On Constrained Intersection Representations of Graphs and Digraphs

Authors Ferdinando Cicalese , Clément Dallard , Martin Milanič

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Ferdinando Cicalese
  • Department of Computer Science, University of Verona, Italy
Clément Dallard
  • Université d'Orléans, INSA Centre Val de Loire, LIFO EA 4022, Orléans, France
Martin Milanič
  • FAMNIT and IAM, University of Primorska, Koper, Slovenia


We would like to thank Andrea Caucchiolo for several insightful discussions we had on some of the results contained in this paper.

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Ferdinando Cicalese, Clément Dallard, and Martin Milanič. On Constrained Intersection Representations of Graphs and Digraphs. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 38:1-38:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


We study the problem of determining minimal directed intersection representations of DAGs in a model introduced by [Kostochka, Liu, Machado, and Milenkovic, ISIT2019]: vertices are assigned color sets, two vertices are connected by an arc if and only if they share at least one color and the tail vertex has a strictly smaller color set than the head, and the goal is to minimize the total number of colors. We show that the problem is polynomially solvable in the class of triangle-free and Hamiltonian DAGs and also disclose the relationship of this problem with several other models of intersection representations of graphs and digraphs.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Mathematics of computing → Graph theory
  • Directed intersection representation
  • intersection number


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