Given a set @math of @math points in the plane, we solve the problems of constructing a geometric planar graph spanning @math 1) of minimum degree 2, and 2) which is 2-edge connected, respectively, and has max edge length bounded by a factor of 2 times the optimal; we also show that the factor 2 is best possible given appropriate connectivity conditions on the set @math , respectively. First, we construct in @math time a geometric planar graph of minimum degree 2 and max edge length bounded by 2 times the optimal. This is then used to construct in @math time a 2-edge connected geometric planar graph spanning @math with max edge length bounded by @math times the optimal, assuming that the set @math forms a connected Unit Disk Graph. Second, we prove that 2 times the optimal is always sufficient if the set of points forms a 2 edge connected Unit Disk Graph and give an algorithm that runs in @math time. We also show that for @math , there exists a set @math of @math points in the plane such that even though the Unit Disk Graph spanning @math is @math -vertex connected, there is no 2-edge connected geometric planar graph spanning @math even if the length of its edges is allowed to be up to 17 16.
T 'oth and Valter @cite_5 characterize geometric planar graphs that can be augmented to 3-edge connected planar graphs. Later Al- @cite_11 gave a tight upper bound on the number of added edges in such augmentations. Finally, Garc ' @cite_8 show how to construct a 3-connected geometric planar graph on a set of points in the planar with the minimum number of straight line edges of unbounded length.
@cite_5 @cite_8 @cite_11
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We characterize the planar straight line graphs (Pslgs) that can be augmented to 3-connected and 3-edge-connected Pslgs, respectively. We show that if a Pslg with n vertices can be augmented to a 3-edge-connected Pslg, then at most 2n−2 new edges are always sufficient and sometimes necessary for the augmentation. If the input Pslg is, in addition, already 2-edge-connected, then n−2 new edges are always sufficient and sometimes necessary for the augmentation to a 3-edge-connected Pslg. It is shown that if a planar straight line graph (Pslg) with n vertices in general position in the plane can be augmented to a 3-edge-connected Pslg, then 2n ? 2 new edges are enough for the augmentation. This bound is tight: there are Pslgs with n ? 4 vertices such that any augmentation to a 3-edge-connected Pslg requires 2n ? 2 new edges. In this paper we study connectivity augmentation problems. Given a connected graph G with some desirable property, we want to make G 2-vertex connected (or 2-edge connected) by adding edges such that the resulting graph keeps the property. The aim is to add as few edges as possible. The property that we consider is planarity, both in an abstract graph-theoretic and in a geometric setting, where vertices correspond to points in the plane and edges to straight-line segments. We show that it is NP-hard to nd a minimum-cardinality augmentation that makes a planar graph 2-edge connected. For making a planar graph 2-vertex connected this was known. We further show that both problems are hard in the geometric setting, even when restricted to trees. The problems remain hard for higher degrees of connectivity. On the other hand we give polynomial-time algorithms for the special case of convex geometric graphs. We also study the following related problem. Given a planar (plane geometric) graph G, two vertices s and t of G, and an integer c, how many edges have to be added to G such that G is still planar (plane geometric) and contains c edge- (or vertex-) disjoint s t paths? For the planar case we give a linear-time algorithm for c = 2. For the plane geometric case we give optimal worst-case bounds for c = 2; for c = 3 we characterize the cases that have a solution. 2-Edge connectivity is an important fault tolerance property of a network because it maintains network communication despite the deletion of a single arbitrary edge. Planar spanning subgraphs have been shown to play a significant role for achieving local decentralized routing in wireless networks. Existing algorithmic constructions of spanning planar subgraphs of unit disk graphs (UDGs) such as Minimum Spanning Tree, Gabriel Graph, Nearest Neighborhood Graph, etc. do not always ensure connectivity of the resulting graph under single edge deletion. Furthermore, adding edges to the network so as to improve its edge connectivity not only may create edge crossings (at points which are not vertices) but it may also require edges of unbounded length. Thus we are faced with the problem of constructing 2-edge connected geometric planar spanning graphs by adding edges of bounded length without creating edge crossings (at points which are not vertices). To overcome this difficulty, in this paper we address the problem of augmenting the edge set (i.e., adding new edges) of planar geometric graphs with straight line edges of bounded length so that the resulting graph is planar and 2-edge connected. We provide bounds on the number of newly added straight-line edges, prove that such edges can be of length at most 3 times the max length of an edge of the original graph, and also show that the factor 3 is optimal. It is shown to be NP-Complete to augment a geometric planar graph to a 2-edge connected geometric planar graph with the minimum number of new edges of a given bounded length. We also provide a constant time algorithm that works in location-aware settings to augment a planar graph into a 2-edge connected planar graph with straight-line edges of length bounded by 3 times the longest edge of the original graph. It turns out that knowledge of vertex coordinates is crucial to our construction and in fact we prove that this problem cannot be solved locally if the vertices do not know their coordinates. Moreover, we provide a family of k-connected UDGs which does not have 2-edge connected spanning planar subgraphs, for any . It is shown that every connected planar straight line graph with n>=3 vertices has an embedding preserving augmentation to a 2-edge connected planar straight line graph with at most @?(2n-2) 3@? new edges. It is also shown that every planar straight line tree with n>=3 vertices has an embedding preserving augmentation to a 2-edge connected planar topological graph with at most @?n 2@? new edges. These bounds are the best possible. However, for every n>=3, there are planar straight line trees with n vertices that do not have an embedding preserving augmentation to a 2-edge connected planar straight line graph with fewer than 1733n-O(1) new edges. We propose a fast methodology for encoding graphs with information-theoretically minimum numbers of bits. Specifically, a graph with property @math is called a @math -graph . If @math satisfies certain properties, then an n-node m-edge @math -graph G can be encoded by a binary string X such that (1) G and X can be obtained from each other in O(n log n) time, and (2) X has at most @math bits for any continuous superadditive function @math so that there are at most @math distinct @math -node @math -graphs. The methodology is applicable to general classes of graphs; this paper focuses on planar graphs. Examples of such @math include all conjunctions over the following groups of properties: (1) G is a planar graph or a plane graph; (2) @math is directed or undirected; (3) @math is triangulated, triconnected, biconnected, merely connected, or not required to be connected; (4) the nodes of G are labeled with labels from @math for @math ; (5) the edges of G are labeled with labels from @math for @math ; and (6) each node (respectively, edge) of G has at most @math self-loops (respectively, @math multiple edges). Moreover, @math and @math are not required to be O(1) for the cases of @math being a plane triangulation. These examples are novel applications of small cycle separators of planar graphs and are the only nontrivial classes of graphs, other than rooted trees, with known polynomial-time information-theoretically optimal coding schemes. It is shown that every plane straight line tree with n vertices can be augmented to a 2-edge connected plane straight line graph by adding at most ⌊n 2⌋ edges, and every connected plane straight line graph with n vertices can be augmented to a 2-edge connected plane straight line graph by adding at most 23n+O(1) edges. Let G be a planar graph with n vertices and with a partition of the vertex set into subsets V"0,...,V"k"-"1 for some positive integer [email protected][email protected]?n. Let S be a set of n distinct points in the plane with a partition into subsets S"0,...,S"k"-"1 with |V"i|=|S"i| ([email protected][email protected]?k-1). This paper studies the problem of computing a planar polyline drawing of G, such that each vertex of V"i is mapped to a distinct point of S"i. Lower and upper bounds on the number of bends per edge are proved for any [email protected][email protected]?n. In the special case k=n, we improve the upper and lower bounds presented in a paper by Pach and Wenger [J. Pach, R. Wenger, Embedding planar graphs at fixed vertex locations, Graphs and Combinatorics 17 (2001) 717-728]. The upper bound is based on an algorithm for computing a topological book embedding of a planar graph, such that the vertices follow a given left-to-right order and the number of crossings between every edge and the spine is asymptotically optimal, which can be regarded as a result of independent interest. Let G be a connected plane geometric graph with n vertices. In this paper, we study bounds on the number of edges required to be added to G to obtain 2-vertex or 2-edge connected plane geometric graphs. In particular, we show that for G to become 2-edge connected, 2n3 additional edges are required in some cases and that 6n7 additional edges are always sufficient. For the special case of plane geometric trees, these bounds decrease to n2 and 2n3, respectively. We give a O(slog n)-approximation algorithm for the sparsest cut, edge expansion, balanced separator, and graph conductance problems. This improves the O(log n)-approximation of Leighton and Rao (1988). We use a well-known semidefinite relaxation with triangle inequality constraints. Central to our analysis is a geometric theorem about projections of point sets in Rd, whose proof makes essential use of a phenomenon called measure concentration. We also describe an interesting and natural “approximate certificate” for a graph's expansion, which involves embedding an n-node expander in it with appropriate dilation and congestion. We call this an expander flow. We give an algorithm to create orthogonal drawings of 3- connected 3-regular planar graphs such that each interior face of the graph is drawn with a prescribed area. This algorithm produces a drawing with at most 12 corners per face and 4 bends per edge, which improves the previous known result of 34 corners per face. Abstract In this paper we describe a linear algorithm for embedding planar graphs in the rectilinear two-dimensional grid, where vertices are grid points and edges are noncrossing grid paths. The main feature of our algorithm is that each edge is guaranteed to have at most 2 bends (with the single exception of the octahedron for which 3 bends are needed). The total number of bends is at most2 n + 4 if the graph is biconnected and at most(7 3) n in the general case. The area is( n + 1) 2 in the worst case. This problem has several applications to VLSI circuit design, aesthetic layout of diagrams, computational geometry. We study the quantitative geometry of graphs in terms of their genus, using the structure of certain "cut graphs," i.e. subgraphs whose removal leaves a planar graph. In particular, we give optimal bounds for random partitioning schemes, as well as various types of embeddings. Using these geometric primitives, we present exponentially improved dependence on genus for a number of problems like approximate max-flow min-cut theorems, approximations for uniform and nonuniform Sparsest Cut, treewidth approximation, Laplacian eigenvalue bounds, and Lipschitz extension theorems and related metric labeling problems. We list here a sample of these improvements. All the following statements refer to graphs of genus g, unless otherwise noted. • We show that such graphs admit an O(log g)-approximate multi-commodity max-flow min-cut theorem for the case of uniform demands. This bound is optimal, and improves over the previous bound of O(g) [KPR93, FT03]. For general demands, we show that the worst possible gap is O(log g + CP), where CP is the gap for planar graphs. This dependence is optimal, and already yields a bound of O(log g + √log n), improving over the previous bound of O(√g log n) [KLMN04]. • We give an O(√log g)-approximation for the uniform Sparsest Cut, balanced vertex separator, and treewidth problems, improving over the previous bound of O(g) [FHL05]. • If a graph G has genus g and maximum degree D, we show that the kth Laplacian eigenvalue of G is (log g)2 · O(kgD n), improving over the previous bound of g2·O(kgD n) [KLPT09]. There is a lower bound of Ω(kgD n), making this result almost tight. • We show that if (X, d) is the shortest-path metric on a graph of genus g and S ⊆ X, then every L-Lipschitz map f: S → Z into a Banach space Z admits an O(L log g)-Lipschitz extension f: X → Z. This improves over the previous bound of O(Lg) [LN05], and compares to a lower bound of Ω(L√log g). In a related way, we show that there is an O(log g)-approximation for the 0-extension problem on such graphs, improving over the previous O(g) bound. • We show that every n-vertex shortest-path metric on a graph of genus g embeds into L2 with distortion O(log g + √log n), improving over the previous bound of O(√g log n). Our result is asymptotically optimal for every dependence g = g(n). Abstract We give new results about the relationship between 1-planar graphs and RAC graphs . A graph is 1-planar if it has a drawing where each edge is crossed at most once. A graph is RAC if it can be drawn in such a way that its edges cross only at right angles. These two classes of graphs and their relationships have been widely investigated in the last years, due to their relevance in application domains where computing readable graph layouts is important to analyze or design relational data sets. We study IC-planar graphs , the sub-family of 1-planar graphs that admit 1-planar drawings with independent crossings (i.e., no two crossed edges share an endpoint). We prove that every IC-planar graph admits a straight-line RAC drawing, which may require however exponential area. If we do not require right angle crossings, we can draw every IC-planar graph with straight-line edges in linear time and quadratic area. We then study the problem of testing whether a graph is IC-planar. We prove that this problem is NP-hard, even if a rotation system for the graph is fixed. On the positive side, we describe a polynomial-time algorithm that tests whether a triangulated plane graph augmented with a given set of edges that form a matching is IC-planar. Abstract Let S be a set of n ⩾ 4 points in general position in the plane, and let h n be the number of extreme points of S . We show how to construct a 3-connected plane graph with vertex set S , having max ⌈ 3 n 2 ⌉ , n + h − 1 edges, and we prove that there is no 3-connected plane graph on top of S with a smaller number of edges. In particular, this implies that S admits a 3-connected cubic plane graph if and only if n ⩾ 4 is even and h ⩽ n 2 + 1 . The same bounds also hold when 3-edge-connectivity is considered. We also give a partial characterization of the point sets in the plane that can be the vertex set of a cubic plane graph. A graph is 1-planar, if it admits a 1-planar embedding, where each edge has at most one crossing. Unfortunately, testing the 1-planarity of a graph is known as NP-complete. This paper initiates the study of the problem of the testing 2-planarity of a graph, in particular, testing the "full-outer-2-planarity" of a graph. A graph is outer-2-planar, if it admits an outer-2-planar embedding, that is every vertex is on the outer boundary and no edge has more than two crossings. A graph is fully-outer-2-planar, if it admits a fully-outer-2-planar embedding, that is an outer-2-planar embedding such that no crossing appears along the outer boundary. We present several structural properties of triconnected outer-2-planar graphs and fully-outer-2-planar graphs, and prove that triconnected fully-outer-2-planar graphs have a constant number of fully-outer-2-planar embeddings. Based on these properties, we present linear-time algorithms for testing the fully-outer-2-planarity of a graph G, whose vertex-connectivity is 1, 2 or at least 3. The algorithm also produces a fully-outer-2-planar embedding of a graph, if it exists. Moreover, we show that every fully-outer-2-planar embedding admits a straight-line drawing. A problem arising in the design of vacuum systems and having applications to some natural problems of interconnection design is described as follows. (1) Given a set X and subsets @math of @math , satisfying @math , find a graph G with vertex set X and the minimum number of edges such that for any i, the subgraph induced by @math has a connected component containing @math .Two other problems related to this one are the following ones. (2) Given a set X and subsets @math such that @math , find a graph G with vertex set X and the minimum number of edges such that for any i the subgraph @math induced by @math in G is connected. (3) Given a set X and subsets @math such that @math , find a graph G with vertex set X, find a graph G with vertex set X and the minimum number of edges such that for any subset I of @math , the subgraph induced by @math is co... This paper addresses the problem of representing the connectivity information of geometric objects, using as little memory as possible. As opposed to raw compression issues, the focus here is on designing data structures that preserve the possibility of answering incidence queries in constant time. We propose, in particular, the first optimal representations for 3-connected planar graphs and triangulations, which are the most standard classes of graphs underlying meshes with spherical topology. Optimal means that these representations asymptotically match the respective entropy of the two classes, namely 2 bits per edge for 3-connected planar graphs, and 1.62 bits per triangle, or equivalently 3.24 bits per vertex for triangulations. These representations support adjacency queries between vertices and faces in constant time. A graph is near-planar if it can be obtained from a planar graph by adding an edge. We show the surprising fact that it is NP-hard to compute the crossing number of near-planar graphs. A graph is 1-planar if it has a drawing where every edge is crossed by at most one other edge. We show that it is NP-hard to decide whether a given near-planar graph is 1-planar. The main idea in both reductions is to consider the problem of simultaneously drawing two planar graphs inside a disk, with some of its vertices fixed at the boundary of the disk. This leads to the concept of anchored embedding, which is of independent interest. As an interesting consequence we obtain a new, geometric proof of NP-completeness of the crossing number problem, even when restricted to cubic graphs. This resolves a question of Hliněný. In this paper, we study the geometric RAC simultaneous drawing problem: Given two planar graphs that share a common vertex set but have disjoint edge sets, a geometric RAC simultaneous drawing is a straight-line drawing in which (i) each graph is drawn planar, (ii) there are no edge overlaps, and, (iii) crossings between edges of the two graphs occur at right-angles. We first prove that two planar graphs admitting a geometric simultaneous drawing may not admit a geometric RAC simultaneous drawing. We further show that a cycle and a matching always admit a geometric RAC simultaneous drawing, which can be constructed in linear time.