21 research outputs found
On the hull and interval numbers of oriented graphs
In this work, for a given oriented graph , we study its interval and hull
numbers, denoted by and , respectively, in the geodetic,
and convexities. This last one, we believe to be formally
defined and first studied in this paper, although its undirected version is
well-known in the literature. Concerning bounds, for a strongly oriented graph
, we prove that and that there is a strongly
oriented graph such that . We also determine exact
values for the hull numbers in these three convexities for tournaments, which
imply polynomial-time algorithms to compute them. These results allows us to
deduce polynomial-time algorithms to compute when the
underlying graph of is split or cobipartite. Moreover, we provide a
meta-theorem by proving that if deciding whether or
is NP-hard or W[i]-hard parameterized by , for some
, then the same holds even if the underlying graph of
is bipartite. Next, we prove that deciding whether or
is W[2]-hard parameterized by , even if the
underlying graph of is bipartite; that deciding whether or is NP-complete, even if has no directed
cycles and the underlying graph of is a chordal bipartite graph; and that
deciding whether or is W[2]-hard
parameterized by , even if the underlying graph of is split. We also
argue that the interval and hull numbers in the oriented and
convexities can be computed in polynomial time for graphs of bounded tree-width
by using Courcelle's theorem
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
On interval number in cycle convexity
International audienceRecently, Araujo et al. [Manuscript in preparation, 2017] introduced the notion of Cycle Convexity of graphs. In their seminal work, they studied the graph convexity parameter called hull number for this new graph convexity they proposed, and they presented some of its applications in Knot theory. Roughly, the tunnel number of a knot embedded in a plane is upper bounded by the hull number of a corresponding planar 4-regular graph in cycle convexity. In this paper, we go further in the study of this new graph convexity and we study the interval number of a graph in cycle convexity. This parameter is, alongside the hull number, one of the most studied parameters in the literature about graph convexities. Precisely, given a graph G, its interval number in cycle convexity, denoted by , is the minimum cardinality of a set S ⊆ V (G) such that every vertex w ∈ V (G) \ S has two distinct neighbors u, v ∈ S such that u and v lie in same connected component of G[S], i.e. the subgraph of G induced by the vertices in S.In this work, first we provide bounds on and its relations to other graph convexity parameters, and explore its behavior on grids. Then, we present some hardness results by showing that deciding whether ≤ k is NP-complete, even if G is a split graph or a bounded-degree planar graph, and that the problem is W[2]-hard in bipartite graphs when k is the parameter. As a consequence, we obtainthat cannot be approximated up to a constant factor in the classes of split graphs and bipartite graphs (unless P = N P ).On the positive side, we present polynomial-time algorithms to compute for outerplanar graphs, cobipartite graphs and interval graphs. We also present fixed-parameter tractable (FPT) algorithms to compute it for (q, q − 4)-graphs when q is the parameter and for general graphs G when parameterized either by the treewidth or the neighborhood diversity of G.Some of our hardness results and positive results are not known to hold for related graph convexities and domination problems. We hope that the design of our new reductions and polynomial-time algorithms can be helpful in order to advance in the study of related graph problems
Independent domination versus packing in subcubic graphs
In 2011, Henning, L\"{o}wenstein, and Rautenbach observed that the domination
number of a graph is bounded from above by the product of the packing number
and the maximum degree of the graph. We prove a stronger statement in subcubic
graphs: the independent domination number is bounded from above by three times
the packing number.Comment: 9 pages, 3 figure
An Introduction to Geometric Topology
This book provides a self-contained introduction to the topology and geometry of surfaces and three-manifolds. The main goal is to describe Thurston's geometrisation of three-manifolds, proved by Perelman in 2002. The book is divided into three parts: the first is devoted to hyperbolic geometry, the second to surfaces, and the third to three-manifolds. It contains complete proofs of Mostow's rigidity, the thick-thin decomposition, Thurston's classification of the diffeomorphisms of surfaces (via Bonahon's geodesic currents), the prime and JSJ decomposition, the topological and geometric classification of Seifert manifolds, and Thurston's hyperbolic Dehn filling Theorem
LIPIcs, Volume 248, ISAAC 2022, Complete Volume
LIPIcs, Volume 248, ISAAC 2022, Complete Volum