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On P3-Convexity of Graphs with Bounded Degree

  • Lucia Draque Penso
  • Fábio Protti
  • Dieter Rautenbach
  • Uéverton S. Souza
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8546)

Abstract

Motivated by the large applicability as well as the hardness of P 3-convexity, we study new complexity aspects of such convexity restricted to graphs with bounded maximum degree. More specifically, we are interested in identifying either a minimum P 3-geodetic set or a minimum P 3-hull set of such graphs, from which the whole vertex set of G is obtained either after one or sufficiently many iterations, respectively. Each iteration adds to a set S all vertices of V(G) ∖ S with at least two neighbors in S. We prove that: (i) a minimum P 3-hull set of a graph G can be found in polynomial time when \(\delta(G)\geq \frac{n(G)}{c}\) (for some constant c); (ii) deciding if the size of a minimum P 3-hull set of a graph is at most k remains NP-complete even on planar graphs with maximum degree four; (iii) a minimum P 3-hull set of a cubic graph can be found in polynomial time; (iv) a minimum P 3-hull set can be found in polynomial time in graphs with minimum feedback vertex set of bounded size and no vertex of degree two; (v) deciding if the size of a minimum P 3-geodetic set of a planar graph with maximum degree three is at most k remains NP-complete.

Keywords

P3-convexity P3-hull set P3-geodetic set planar graphs bounded degree NP-hardness 

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Lucia Draque Penso
    • 1
  • Fábio Protti
    • 2
  • Dieter Rautenbach
    • 1
  • Uéverton S. Souza
    • 2
  1. 1.Universität UlmUlmGermany
  2. 2.IC - Universidade Federal FluminenseNiteróiBrazil

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