The Geodetic Hull Number is Hard for Chordal Graphs

We show the hardness of the geodetic hull number for chordal graphs

The general position number and the iteration time in the P3 convexity

In this paper, we investigate two graph convexity parameters: the iteration time and the general position number. Harary and Nieminem introduced in 1981 the iteration time in the geodesic convexity, but its computational complexity was still open. Manuel and Klav\v{z}ar introduced in 2018 the general position number of the geodesic convexity and proved that it is NP-hard to compute. In this paper, we extend these parameters to the P3 convexity and prove that it is NP-hard to compute them. With this, we also prove that the iteration number is NP-hard on the geodesic convexity even in graphs with diameter two. These results are the last three missing NP-hardness results regarding the ten most studied graph convexity parameters in the geodesic and P3 convexities

The colorful Helly theorem and general hypergraphs

AbstractThe definition of the Helly property for hypergraphs was motivated by the Helly theorem for convex sets. Similarly, we define the colorful Helly property for a family of hypergraphs, motivated by the colorful Helly theorem for collections of convex sets, by Lovász. We describe some general facts about the colorful Helly property and prove complexity results. In particular, we show that it is Co-NP-complete to decide if a family of p hypergraphs is colorful Helly, even if p=2. However, for any fixed p, we describe a polynomial time algorithm to decide if such family is colorful Helly, provided at least p−1 of the hypergraphs are p-Helly

Mixed unit interval graphs

AbstractThe class of intersection graphs of unit intervals of the real line whose ends may be open or closed is a strict superclass of the well-known class of unit interval graphs. We pose a conjecture concerning characterizations of such mixed unit interval graphs, verify parts of it in general, and prove it completely for diamond-free graphs. In particular, we characterize diamond-free mixed unit interval graphs by means of an infinite family of forbidden induced subgraphs, and we show that a diamond-free graph is mixed unit interval if and only if it has intersection representations using unit intervals such that all ends of the intervals are integral

On the hull number of triangle-free graphs

A set of vertices C in a graph is convex if it contains all vertices which lie on shortest paths between vertices in C. The convex hull of a set of vertices S is the smallest convex set containing S. The hull number h(G) of a graph G is the smallest cardinality of a set of vertices whose convex hull is the vertex set of G. For a connected triangle-free graph G of order n and diameter d\geq 3, we prove that h(G)\leq (n-d+3)/3, if G has minimum degree at least 3 and that h(G)\leq 2(n-d+5)/7, if G is cubic. Furthermore, for a connected graph G of order n, girth g\geq 4, minimum degree at least 2, and diameter d, we prove h(G)\leq 2+ (n-d-1)/\left\lceil\frac{g-1}{2}\right \rceil. All bounds are best possible

Some remarks on the geodetic number of a graph

A set of vertices D of a graph G is geodetic if every vertex of G lies on a shortest path between two not necessarily distinct vertices in D. The geodetic number of G is the minimum cardinality of a geodetic set of G. We prove that it is NP complete to decide for a given chordal or chordal bipartite graph G and a given integer k whether G has a geodetic set of cardinality at most k. Furthermore, we prove an upper bound on the geodetic number of graphs without short cycles and study the geodetic number of cographs, split graphs, and unit interval graphs

On the convexity number of graphs

A set of vertices S in a graph is convex if it contains all vertices which belong to shortest paths between vertices in S. The convexity number c(G) of a graph G is the maximum cardinality of a convex set of vertices which does not contain all vertices of G. We prove NP-completeness of the problem to decide for a given bipartite graph G and an integer k whether c(G)\geq k. Furthermore, we identify natural necessary extension properties of graphs of small convexity number and study the interplay between these properties and upper bounds on the convexity number