21 research outputs found

    On the hull and interval numbers of oriented graphs

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    In this work, for a given oriented graph DD, we study its interval and hull numbers, denoted by in(D){in}(D) and hn(D){hn}(D), respectively, in the geodetic, P3{P_3} and P3∗{P_3^*} 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 DD, we prove that hng(D)≤m(D)−n(D)+2{hn_g}(D)\leq m(D)-n(D)+2 and that there is a strongly oriented graph such that hng(D)=m(D)−n(D){hn_g}(D) = m(D)-n(D). 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 hnP3(D){hn_{P_3}}(D) when the underlying graph of DD is split or cobipartite. Moreover, we provide a meta-theorem by proving that if deciding whether ing(D)≤k{in_g}(D)\leq k or hng(D)≤k{hn_g}(D)\leq k is NP-hard or W[i]-hard parameterized by kk, for some i∈Z+∗i\in\mathbb{Z_+^*}, then the same holds even if the underlying graph of DD is bipartite. Next, we prove that deciding whether hnP3(D)≤k{hn_{P_3}}(D)\leq k or hnP3∗(D)≤k{hn_{P_3^*}}(D)\leq k is W[2]-hard parameterized by kk, even if the underlying graph of DD is bipartite; that deciding whether inP3(D)≤k{in_{P_3}}(D)\leq k or inP3∗(D)≤k{in_{P_3^*}}(D)\leq k is NP-complete, even if DD has no directed cycles and the underlying graph of DD is a chordal bipartite graph; and that deciding whether inP3(D)≤k{in_{P_3}}(D)\leq k or inP3∗(D)≤k{in_{P_3^*}}(D)\leq k is W[2]-hard parameterized by kk, even if the underlying graph of DD is split. We also argue that the interval and hull numbers in the oriented P3P_3 and P3∗P_3^* 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

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    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 u,v∈Vu,v\in V of a connected graph G=(V,E)G=(V,E), the {\em closed interval} I[u,v]I[u,v] of uu and vv is the the set of vertices that belong to some shortest (u,v)(u,v)-path. For any S⊆VS \subseteq V, let I[S]=⋃u,v∈SI[u,v]I[S]= \bigcup_{u,v\in S} I[u,v]. A subset S⊆VS\subseteq V is {\em geodesically convex} if I[S]=SI[S] = S. In other words, a subset SS is convex if, for any u,v∈Su,v \in S and for any shortest (u,v)(u,v)-path PP, V(P)⊆SV(P) \subseteq S. Given a subset S⊆VS\subseteq V, the {\em convex hull} Ih[S]I_h[S] of SS is the smallest convex set that contains SS. We say that SS is a {\em hull set} of GG if Ih[S]=VI_h[S] = V. The size of a minimum hull set of GG is the {\em hull number} of GG, denoted by hn(G)hn(G). The {\sc Hull Number} problem is to decide whether hn(G)≤khn(G)\leq k, for a given graph GG and an integer kk. 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 (q,q−4)(q,q-4)-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 nn-node graph GG without simplicial vertices is at most 1+⌈3(n−1)5⌉1+\lceil \frac{3(n-1)}{5}\rceil in general, at most 1+⌈n−12⌉1+\lceil \frac{n-1}{2}\rceil if GG is regular or has no triangle, and at most 1+⌈n−13⌉1+\lceil \frac{n-1}{3}\rceil if GG has girth at least 66.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 G=(V,E)G = (V,E), l'intervalle I[u,v]I[u,v] entre deux sommets u,v∈Vu,v \in V est l'ensemble des sommets qui appartiennent à un plus court chemin entre uu et vv. Pour un ensemble S⊆VS\subseteq V, on note I[S]I[S] l'ensemble ⋃u,v∈SI[u,v]\bigcup_{u,v\in S} I[u,v]. Un ensemble S⊆VS\subseteq V de sommets est dit {\it convexe} si I[S]=SI[S] = S. L'{\it enveloppe convexe} Ih[S]I_h[S] d'un sous-ensemble S⊆VS\subseteq V de GG est défini comme le plus petit ensemble convexe qui contient SS. S⊆VS\subseteq V est une {\it enveloppe} de GG si Ih[S]=VI_h[S] = V. Le {\it nombre enveloppe} de GG, noté hn(G)hn(G), est la cardinalité minimum d'une enveloppe de graphe GG. Nous montrons que décider si hn(G)≤khn(G) \leq k est un problème NP-complet dans la classe des graphes bipartis et nous prouvons que hn(G)hn(G) peut être calculé en temps polynomial pour les cobipartis, (q,q−4)(q,q-4)-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

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    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 u,v∈Vu,v\in V of a connected graph G=(V,E)G=(V,E), the closed interval I[u,v]I[u,v] of uu and vv is the the set of vertices that belong to some shortest (u,v)(u,v)-path. For any S⊆VS \subseteq V, let I[S]=⋃u,v∈SI[u,v]I[S]= \bigcup_{u,v\in S} I[u,v]. A subset S⊆VS\subseteq V is geodesically convex or convex if I[S]=SI[S] = S. In other words, a subset SS is convex if, for any u,v∈Su,v \in S and for any shortest (u,v)(u,v)-path PP, V(P)⊆SV(P) \subseteq S. Given a subset S⊆VS\subseteq V, the convex hull Ih[S]I_h[S] of SS is the smallest convex set that contains SS. We say that SS is a hull set of GG if Ih[S]=VI_h[S] = V. The size of a minimum hull set of GG is the hull number of GG, denoted by hn(G)hn(G). The {\sc Hull Number} problem is to decide whether hn(G)≤khn(G)\leq k, for a given graph GG and an integer kk. 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 (q,q−4)(q,q-4)-graphs and to cacti. Finally, we prove tight upper bounds on the hull numbers. In particular, we show that the hull number of an nn-node graph GG without simplicial vertices is at most 1+⌈3(n−1)5⌉1+\lceil \frac{3(n-1)}{5}\rceil in general, at most 1+⌈n−12⌉1+\lceil \frac{n-1}{2}\rceil if GG is regular or has no triangle, and at most 1+⌈n−13⌉1+\lceil \frac{n-1}{3}\rceil if GG has girth at least 66

    On interval number in cycle convexity

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    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 incc(G)in_{cc} (G), 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 incc(G)in_{cc} (G) 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 incc(G)in_{cc} (G) ≤ 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 incc(G)in_{cc} (G) 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 incc(G)in_{cc} (G) 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

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    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

    Subject Index Volumes 1–200

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    An Introduction to Geometric Topology

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    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

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    LIPIcs, Volume 248, ISAAC 2022, Complete Volum
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