On P ₃-Convexity of Graphs with Bounded Degree

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.