1,966 research outputs found

    Sufficient Conditions for Tuza's Conjecture on Packing and Covering Triangles

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    Given a simple graph G=(V,E)G=(V,E), a subset of EE is called a triangle cover if it intersects each triangle of GG. Let νt(G)\nu_t(G) and τt(G)\tau_t(G) denote the maximum number of pairwise edge-disjoint triangles in GG and the minimum cardinality of a triangle cover of GG, respectively. Tuza conjectured in 1981 that τt(G)/νt(G)≤2\tau_t(G)/\nu_t(G)\le2 holds for every graph GG. In this paper, using a hypergraph approach, we design polynomial-time combinatorial algorithms for finding small triangle covers. These algorithms imply new sufficient conditions for Tuza's conjecture on covering and packing triangles. More precisely, suppose that the set TG\mathscr T_G of triangles covers all edges in GG. We show that a triangle cover of GG with cardinality at most 2νt(G)2\nu_t(G) can be found in polynomial time if one of the following conditions is satisfied: (i) νt(G)/∣TG∣≥13\nu_t(G)/|\mathscr T_G|\ge\frac13, (ii) νt(G)/∣E∣≥14\nu_t(G)/|E|\ge\frac14, (iii) ∣E∣/∣TG∣≥2|E|/|\mathscr T_G|\ge2. Keywords: Triangle cover, Triangle packing, Linear 3-uniform hypergraphs, Combinatorial algorithm

    Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs

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    We study the family of intersection graphs of low density objects in low dimensional Euclidean space. This family is quite general, and includes planar graphs. We prove that such graphs have small separators. Next, we present efficient (1+ε)(1+\varepsilon)-approximation algorithms for these graphs, for Independent Set, Set Cover, and Dominating Set problems, among others. We also prove corresponding hardness of approximation for some of these optimization problems, providing a characterization of their intractability in terms of density

    A Pfaffian formula for monomer-dimer partition functions

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    We consider the monomer-dimer partition function on arbitrary finite planar graphs and arbitrary monomer and dimer weights, with the restriction that the only non-zero monomer weights are those on the boundary. We prove a Pfaffian formula for the corresponding partition function. As a consequence of this result, multipoint boundary monomer correlation functions at close packing are shown to satisfy fermionic statistics. Our proof is based on the celebrated Kasteleyn theorem, combined with a theorem on Pfaffians proved by one of the authors, and a careful labeling and directing procedure of the vertices and edges of the graph.Comment: Added referenc

    On the tractability of some natural packing, covering and partitioning problems

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    In this paper we fix 7 types of undirected graphs: paths, paths with prescribed endvertices, circuits, forests, spanning trees, (not necessarily spanning) trees and cuts. Given an undirected graph G=(V,E)G=(V,E) and two "object types" A\mathrm{A} and B\mathrm{B} chosen from the alternatives above, we consider the following questions. \textbf{Packing problem:} can we find an object of type A\mathrm{A} and one of type B\mathrm{B} in the edge set EE of GG, so that they are edge-disjoint? \textbf{Partitioning problem:} can we partition EE into an object of type A\mathrm{A} and one of type B\mathrm{B}? \textbf{Covering problem:} can we cover EE with an object of type A\mathrm{A}, and an object of type B\mathrm{B}? This framework includes 44 natural graph theoretic questions. Some of these problems were well-known before, for example covering the edge-set of a graph with two spanning trees, or finding an ss-tt path PP and an s′s'-t′t' path P′P' that are edge-disjoint. However, many others were not, for example can we find an ss-tt path P⊆EP\subseteq E and a spanning tree T⊆ET\subseteq E that are edge-disjoint? Most of these previously unknown problems turned out to be NP-complete, many of them even in planar graphs. This paper determines the status of these 44 problems. For the NP-complete problems we also investigate the planar version, for the polynomial problems we consider the matroidal generalization (wherever this makes sense)

    Decomposition of multiple packings with subquadratic union complexity

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    Suppose kk is a positive integer and X\mathcal{X} is a kk-fold packing of the plane by infinitely many arc-connected compact sets, which means that every point of the plane belongs to at most kk sets. Suppose there is a function f(n)=o(n2)f(n)=o(n^2) with the property that any nn members of X\mathcal{X} determine at most f(n)f(n) holes, which means that the complement of their union has at most f(n)f(n) bounded connected components. We use tools from extremal graph theory and the topological Helly theorem to prove that X\mathcal{X} can be decomposed into at most pp (11-fold) packings, where pp is a constant depending only on kk and ff.Comment: Small generalization of the main result, improvements in the proofs, minor correction

    Nested hierarchies in planar graphs

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    We construct a partial order relation which acts on the set of 3-cliques of a maximal planar graph G and defines a unique hierarchy. We demonstrate that G is the union of a set of special subgraphs, named `bubbles', that are themselves maximal planar graphs. The graph G is retrieved by connecting these bubbles in a tree structure where neighboring bubbles are joined together by a 3-clique. Bubbles naturally provide the subdivision of G into communities and the tree structure defines the hierarchical relations between these communities
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