On the hull number on cycle convexity of graphs

In this work, we study the parameter hull number in a recently defined graph convexity called Cycle Convexity, whose definition is motivated by related notions in Knot Theory. For a graph G=(V,E), define the interval function in the Cycle Convexity as Icc(S)=S∪{v∈V(G)|there is a cycle C in G such that V(C)∖S={v}}, for every S⊆V(G). We say that S⊆V(G) is convex if Icc(S)=S. The convex hull of S⊆V(G), denoted by Hull(S), is the inclusion-wise minimal convex set S′ such that S⊆S′. A set S⊆V(G) is called a hull set if Hull(S)=V(G). The hull number of G in the cycle convexity, denoted by hncc(G), is the cardinality of a smallest hull set of G. We first focus on the class of planar graphs, as the main motivation for the definition of hncc(G) stems from Knot Theory and occurs when G is a 4-regular planar graph. We prove that: the hull number of a 4-regular planar graph is at most half of its number of vertices and that such bound is tight; and that deciding whether hncc(G)≤k, provided a positive integer k and a planar graph G, is an NP-complete problem. On the positive side, we present polynomial-time algorithms to compute the hull number in the cycle convexity of chordal graphs, P4-sparse graphs, and grids.