Computing metric hulls in graphs

We prove that, given a closure function the smallest preimage of a closed set can be calculated in polynomial time in the number of closed sets. This confirms a conjecture of Albenque and Knauer and implies that there is a polynomial time algorithm to compute the convex hull-number of a graph, when all its convex subgraphs are given as input. We then show that computing if the smallest preimage of a closed set is logarithmic in the size of the ground set is LOGSNP-complete if only the ground set is given. A special instance of this problem is computing the dimension of a poset given its linear extension graph, that was conjectured to be in P. The intent to show that the latter problem is LOGSNP-complete leads to several interesting questions and to the definition of the isometric hull, i.e., a smallest isometric subgraph containing a given set of vertices $S$. While for $|S|=2$ an isometric hull is just a shortest path, we show that computing the isometric hull of a set of vertices is NP-complete even if $|S|=3$. Finally, we consider the problem of computing the isometric hull-number of a graph and show that computing it is $\Sigma^P_2$ complete.Comment: 13 pages, 3 figure

The forcing hull and forcing geodetic numbers of graphs

AbstractFor every pair of vertices u,v in a graph, a u–v geodesic is a shortest path from u to v. For a graph G, let IG[u,v] denote the set of all vertices lying on a u–v geodesic. Let S⊆V(G) and IG[S] denote the union of all IG[u,v] for all u,v∈S. A subset S⊆V(G) is a convex set of G if IG[S]=S. A convex hull [S]G of S is a minimum convex set containing S. A subset S of V(G) is a hull set of G if [S]G=V(G). The hull number h(G) of a graph G is the minimum cardinality of a hull set in G. A subset S of V(G) is a geodetic set if IG[S]=V(G). The geodetic number g(G) of a graph G is the minimum cardinality of a geodetic set in G. A subset F⊆V(G) is called a forcing hull (or geodetic) subset of G if there exists a unique minimum hull (or geodetic) set containing F. The cardinality of a minimum forcing hull subset in G is called the forcing hull number fh(G) of G and the cardinality of a minimum forcing geodetic subset in G is called the forcing geodetic number fg(G) of G. In the paper, we construct some 2-connected graph G with (fh(G),fg(G))=(0,0),(1,0), or (0,1), and prove that, for any nonnegative integers a, b, and c with a+b≥2, there exists a 2-connected graph G with (fh(G),fg(G),h(G),g(G))=(a,b,a+b+c,a+2b+c) or (a,2a+b,a+b+c,2a+2b+c). These results confirm a conjecture of Chartrand and Zhang proposed in [G. Chartrand, P. Zhang, The forcing hull number of a graph, J. Combin. Math. Combin. Comput. 36 (2001) 81–94]

Convexity in partial cubes: the hull number

We prove that the combinatorial optimization problem of determining the hull number of a partial cube is NP-complete. This makes partial cubes the minimal graph class for which NP-completeness of this problem is known and improves some earlier results in the literature. On the other hand we provide a polynomial-time algorithm to determine the hull number of planar partial cube quadrangulations. Instances of the hull number problem for partial cubes described include poset dimension and hitting sets for interiors of curves in the plane. To obtain the above results, we investigate convexity in partial cubes and characterize these graphs in terms of their lattice of convex subgraphs, improving a theorem of Handa. Furthermore we provide a topological representation theorem for planar partial cubes, generalizing a result of Fukuda and Handa about rank three oriented matroids.Comment: 19 pages, 4 figure

On the geodesic pre-hull number of a graph

AbstractGiven a convexity space X whose structure is induced by an interval operator I, we define a parameter, called the pre-hull number of X, which measures the intrinsic non-convexity of X in terms of the number of iterations of the pre-hull operator associated with I which are necessary in the worst case to reach the canonical extension of copoints of X when they are being extended by the adjunction of an attaching point. We consider primarily the geodesic convexity structure of connected graphs in the case where the pre-hull number is at most 1, with emphasis on bipartite graphs, in particular, partial cubes

Computing the hull number in toll convexity

A walk W between vertices u and v of a graph G is called a tolled walk between u and v if u, as well as v, has exactly one neighbour in W. A set S ⊆ V (G) is toll convex if the vertices contained in any tolled walk between two vertices of S are contained in S. The toll convex hull of S is the minimum toll convex set containing S. The toll hull number of G is the minimum cardinality of a set S such that the toll convex hull of S is V (G). The main contribution of this work is an algorithm for computing the toll hull number of a general graph in polynomial time