16 research outputs found
On the hull number of some graph classes
In this paper, we study the geodetic convexity of graphs focusing on the problem of the complexity to compute inclusion-minimum hull set of a graph in several graph classes. For any two vertices of a connected graph , the {\em closed interval} of and is the the set of vertices that belong to some shortest -path. For any , let . A subset is {\em geodesically convex} if . In other words, a subset is convex if, for any and for any shortest -path , . Given a subset , the {\em convex hull} of is the smallest convex set that contains . We say that is a {\em hull set} of if . The size of a minimum hull set of is the {\em hull number} of , denoted by . The {\sc Hull Number} problem is to decide whether , for a given graph and an integer . Dourado {\it et al.} showed that this problem is NP-complete in general graphs. In this paper, we answer an open question of Dourado et al.~\cite{Douradoetal09} by showing that the {\sc Hull Number} problem is NP-hard even when restricted to the class of bipartite graphs. Then, we design polynomial time algorithms to solve the {\sc Hull Number} problem in several graph classes. First, we deal with the class of complements of bipartite graphs. Then, we generalize some results in~\cite{ACGSS11} to the class of -graphs and to the class of cacti. Finally, we prove tight upper bounds on the hull numbers. In particular, we show that the hull number of an -node graph without simplicial vertices is at most in general, at most if is regular or has no triangle, and at most if has girth at least .Dans cet article nous étudions une notion de convexité dans les graphes. Nous nous concentrons sur la question de la compléxité du calcul de l'enveloppe minimum d'un graphe dans le cas de diverses classes de graphes. Étant donné un graphe , l'intervalle entre deux sommets est l'ensemble des sommets qui appartiennent à un plus court chemin entre et . Pour un ensemble , on note l'ensemble . Un ensemble de sommets est dit {\it convexe} si . L'{\it enveloppe convexe} d'un sous-ensemble de est défini comme le plus petit ensemble convexe qui contient . est une {\it enveloppe} de si . Le {\it nombre enveloppe} de , noté , est la cardinalité minimum d'une enveloppe de graphe . Nous montrons que décider si est un problème NP-complet dans la classe des graphes bipartis et nous prouvons que peut être calculé en temps polynomial pour les cobipartis, -graphes et cactus. Nous montrons aussi des bornes supérieures du nombre enveloppe des graphes en général, des graphes sans triangles et des graphes réguliers
On the hull number of some graph classes
International audienceIn this paper, we study the geodetic convexity of graphs focusing on the problem of the complexity to compute a minimum hull set of a graph in several graph classes. For any two vertices of a connected graph , the closed interval of and is the the set of vertices that belong to some shortest -path. For any , let . A subset is geodesically convex or convex if . In other words, a subset is convex if, for any and for any shortest -path , . Given a subset , the convex hull of is the smallest convex set that contains . We say that is a hull set of if . The size of a minimum hull set of is the hull number of , denoted by . The {\sc Hull Number} problem is to decide whether , for a given graph and an integer . Dourado {\it et al.} showed that this problem is NP-complete in general graphs. In this paper, we answer an open question of Dourado {\it et al.}~\cite{Douradoetal09} by showing that the {\sc Hull Number} problem is NP-hard even when restricted to the class of bipartite graphs. Then, we design polynomial time algorithms to solve the {\sc Hull Number} problem in several graph classes. First, we deal with the class of complements of bipartite graphs. Then, we generalize some results in~\cite{ACGSS11} to the class of -graphs and to cacti. Finally, we prove tight upper bounds on the hull numbers. In particular, we show that the hull number of an -node graph without simplicial vertices is at most in general, at most if is regular or has no triangle, and at most if has girth at least
The Geodetic Hull Number is Hard for Chordal Graphs
We show the hardness of the geodetic hull number for chordal graphs
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
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 .
While for 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
. Finally, we consider the problem of computing the isometric
hull-number of a graph and show that computing it is complete.Comment: 13 pages, 3 figure
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
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