The Graph Tessellation Cover Number: Extremal Bounds, Efficient Algorithms and Hardness

  • Alexandre Abreu
  • Luís Cunha
  • Tharso Fernandes
  • Celina de Figueiredo
  • Luis Kowada
  • Franklin Marquezino
  • Daniel Posner
  • Renato Portugal
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10807)

Abstract

A tessellation of a graph is a partition of its vertices into vertex disjoint cliques. A tessellation cover of a graph is a set of tessellations that covers all of its edges. The t-tessellability problem aims to decide whether there is a tessellation cover of the graph with t tessellations. This problem is motivated by its applications to quantum walk models, in especial, the evolution operator of the staggered model is obtained from a graph tessellation cover. We establish upper bounds on the tessellation cover number given by the minimum between the chromatic index of the graph and the chromatic number of its clique graph and we show graph classes for which these bounds are tight. We prove \(\mathcal {NP}\)-completeness for t-tessellability if the instance is restricted to planar graphs, chordal (2, 1)-graphs, (1, 2)-graphs, diamond-free graphs with diameter five, or for any fixed t at least 3. On the other hand, we improve the complexity for 2-tessellability to a linear-time algorithm.

Keywords

Staggered quantum walk Clique graph Tessellation 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Alexandre Abreu
    • 1
  • Luís Cunha
    • 2
  • Tharso Fernandes
    • 3
    • 4
  • Celina de Figueiredo
    • 1
  • Luis Kowada
    • 2
  • Franklin Marquezino
    • 1
  • Daniel Posner
    • 1
  • Renato Portugal
    • 4
  1. 1.Universidade Federal do Rio de JaneiroRio de JaneiroBrazil
  2. 2.Universidade Federal FluminenseNiteróiBrazil
  3. 3.Universidade Federal do Espírito SantoVitóriaBrazil
  4. 4.Laboratório Nacional de Computação CientíficaPetrópolisBrazil

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