48,781 research outputs found

    The Minimum Shared Edges Problem on Grid-like Graphs

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    We study the NP-hard Minimum Shared Edges (MSE) problem on graphs: decide whether it is possible to route pp paths from a start vertex to a target vertex in a given graph while using at most kk edges more than once. We show that MSE can be decided on bounded (i.e. finite) grids in linear time when both dimensions are either small or large compared to the number pp of paths. On the contrary, we show that MSE remains NP-hard on subgraphs of bounded grids. Finally, we study MSE from a parametrised complexity point of view. It is known that MSE is fixed-parameter tractable with respect to the number pp of paths. We show that, under standard complexity-theoretical assumptions, the problem parametrised by the combined parameter kk, pp, maximum degree, diameter, and treewidth does not admit a polynomial-size problem kernel, even when restricted to planar graphs

    On the diameter of random planar graphs

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    We show that the diameter D(G_n) of a random labelled connected planar graph with n vertices is equal to n^{1/4+o(1)}, in probability. More precisely there exists a constant c>0 such that the probability that D(G_n) lies in the interval (n^{1/4-\epsilon},n^{1/4+\epsilon}) is greater than 1-\exp(-n^{c\epsilon}) for {\epsilon} small enough and n>n_0(\epsilon). We prove similar statements for 2-connected and 3-connected planar graphs and maps.Comment: 24 pages, 7 figure

    Diameter and Treewidth in Minor-Closed Graph Families

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    It is known that any planar graph with diameter D has treewidth O(D), and this fact has been used as the basis for several planar graph algorithms. We investigate the extent to which similar relations hold in other graph families. We show that treewidth is bounded by a function of the diameter in a minor-closed family, if and only if some apex graph does not belong to the family. In particular, the O(D) bound above can be extended to bounded-genus graphs. As a consequence, we extend several approximation algorithms and exact subgraph isomorphism algorithms from planar graphs to other graph families.Comment: 15 pages, 12 figure

    The degree/diameter problem in maximal planar bipartite graphs

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    The (¿;D) (degree/diameter) problem consists of nding the largest possible number of vertices n among all the graphs with maximum degree ¿ and diameter D. We consider the (¿;D) problem for maximal planar bipartite graphs, that are simple planar graphs in which every face is a quadrangle. We obtain that for the (¿; 2) problem, the number of vertices is n = ¿+2; and for the (¿; 3) problem, n = 3¿¿1 if ¿ is odd and n = 3¿ ¿ 2 if ¿ is even. Then, we study the general case (¿;D) and obtain that an upper bound on n is approximately 3(2D + 1)(¿ ¿ 2)¿D=2¿ and another one is C(¿ ¿ 2)¿D=2¿ if ¿ D and C is a sufficiently large constant. Our upper bound improve for our kind of graphs the one given by Fellows, Hell and Seyffarth for general planar graphs. We also give a lower bound on n for maximal planar bipartite graphs, which is approximately (¿ ¿ 2)k if D = 2k, and 3(¿ ¿ 3)k if D = 2k + 1, for ¿ and D sufficiently large in both cases.Postprint (published version

    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

    Vertex-Coloring with Star-Defects

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    Defective coloring is a variant of traditional vertex-coloring, according to which adjacent vertices are allowed to have the same color, as long as the monochromatic components induced by the corresponding edges have a certain structure. Due to its important applications, as for example in the bipartisation of graphs, this type of coloring has been extensively studied, mainly with respect to the size, degree, and acyclicity of the monochromatic components. In this paper we focus on defective colorings in which the monochromatic components are acyclic and have small diameter, namely, they form stars. For outerplanar graphs, we give a linear-time algorithm to decide if such a defective coloring exists with two colors and, in the positive case, to construct one. Also, we prove that an outerpath (i.e., an outerplanar graph whose weak-dual is a path) always admits such a two-coloring. Finally, we present NP-completeness results for non-planar and planar graphs of bounded degree for the cases of two and three colors

    Diameter Bounds for Planar Graphs

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    The inverse degree of a graph is the sum of the reciprocals of the degrees of its vertices. We prove that in any connected planar graph, the diameter is at most 5/2 times the inverse degree, and that this ratio is tight. To develop a crucial surgery method, we begin by proving the simpler related upper bounds (4(V-1)-E)/3 and 4V^2/3E on the diameter (for connected planar graphs), which are also tight

    On the maximum order of graphs embedded in surfaces

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    The maximum number of vertices in a graph of maximum degree Δ≥3\Delta\ge 3 and fixed diameter k≥2k\ge 2 is upper bounded by (1+o(1))(Δ−1)k(1+o(1))(\Delta-1)^{k}. If we restrict our graphs to certain classes, better upper bounds are known. For instance, for the class of trees there is an upper bound of (2+o(1))(Δ−1)⌊k/2⌋(2+o(1))(\Delta-1)^{\lfloor k/2\rfloor} for a fixed kk. The main result of this paper is that graphs embedded in surfaces of bounded Euler genus gg behave like trees, in the sense that, for large Δ\Delta, such graphs have orders bounded from above by begin{cases} c(g+1)(\Delta-1)^{\lfloor k/2\rfloor} & \text{if $k$ is even} c(g^{3/2}+1)(\Delta-1)^{\lfloor k/2\rfloor} & \text{if $k$ is odd}, \{cases} where cc is an absolute constant. This result represents a qualitative improvement over all previous results, even for planar graphs of odd diameter kk. With respect to lower bounds, we construct graphs of Euler genus gg, odd diameter kk, and order c(g+1)(Δ−1)⌊k/2⌋c(\sqrt{g}+1)(\Delta-1)^{\lfloor k/2\rfloor} for some absolute constant c>0c>0. Our results answer in the negative a question of Miller and \v{S}ir\'a\v{n} (2005).Comment: 13 pages, 3 figure

    The degree-diameter problem for sparse graph classes

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    The degree-diameter problem asks for the maximum number of vertices in a graph with maximum degree Δ\Delta and diameter kk. For fixed kk, the answer is Θ(Δk)\Theta(\Delta^k). We consider the degree-diameter problem for particular classes of sparse graphs, and establish the following results. For graphs of bounded average degree the answer is Θ(Δk−1)\Theta(\Delta^{k-1}), and for graphs of bounded arboricity the answer is \Theta(\Delta^{\floor{k/2}}), in both cases for fixed kk. For graphs of given treewidth, we determine the the maximum number of vertices up to a constant factor. More precise bounds are given for graphs of given treewidth, graphs embeddable on a given surface, and apex-minor-free graphs
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