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.
@cite_0 give a polynomial algorithm which augments any geometric planar graph to 2-vertex connected or 2-edge connected geometric planar graph, respectively, but no bounds are given on the length of the augmented edges. T 'oth @cite_6 improves the bound on the number of necessary edges in such augmentations, and Rutter and Wolff @cite_7 prove that it is NP-hard to determine the minimum number of edges that have to be added in such augmentations.
@cite_0 @cite_7 @cite_6
2063013286 2098823334 2046895810
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. 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. Given a connected geometric graph G, we consider the problem of constructing a t-spanner of G having the minimum number of edges. We prove that for every t with @math , there exists a connected geometric graph G with n vertices, such that every t-spanner of G contains Ω( n1+1 t ) edges. This bound almost matches the known upper bound, which states that every connected weighted graph with n vertices contains a t-spanner with O(tn1+2 (t+1)) edges. We also prove that the problem of deciding whether a given geometric graph contains a t-spanner with at most K edges is NP-hard. Previously, this NP-hardness result was only known for non-geometric graphs 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. 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. Being motivated by John Tantalo's Planarity Game, we consider straight line plane drawings of a planar graph G with edge crossings and wonder how obfuscated such drawings can be. We define obf(G), the obfuscation complexity of G, to be the maximum number of edge crossings in a drawing of G. Relating obf(G) to the distribution of vertex degrees in G, we show an efficient way of constructing a drawing of G with at least obf(G) 3 edge crossings. We prove bounds (@d(G)^2 24-o(1))n^2@?obf(G) =2. The shift complexity of G, denoted by shift(G), is the minimum number of vertex shifts sufficient to eliminate all edge crossings in an arbitrarily obfuscated drawing of G (after shifting a vertex, all incident edges are supposed to be redrawn correspondingly). If @d(G)>=3, then shift(G) is linear in the number of vertices due to the known fact that the matching number of G is linear. However, in the case @d(G)>=2 we notice that shift(G) can be linear even if the matching number is bounded. As for computational complexity, we show that, given a drawing D of a planar graph, it is NP-hard to find an optimum sequence of shifts making D crossing-free. In the survivable network design problem (SNDP), the goal is to find a minimum-cost spanning subgraph satisfying certain connectivity requirements. We study the vertex-connectivity variant of SNDP in which the input specifies, for each pair of vertices, a required number of vertex-disjoint paths connecting them. We give the first strong lower bound on the approximability of SNDP, showing that the problem admits no efficient @math ratio approximation for any fixed @math , unless @math . We show hardness of approximation results for some important special cases of SNDP, and we exhibit the first lower bound on the approximability of the related classical NP-hard problem of augmenting the connectivity of a graph using edges from a given set. A subset T@?V of terminals is k-connected to a root s in a directed undirected graph J if J has k internally-disjoint vs-paths for every v@?T; T is k-connected in J if for every s@?T the set T@? s is k-connected to s in J. We consider the Subsetk-ConnectivityAugmentation problem: given a graph G=(V,E) with edge node-costs, a node subset T@?V, and a subgraph J=(V,E"J) of G such that T is (k-1)-connected in J, find a minimum-cost augmenting edge-set F@?E@?E"J such that T is k-connected in J@?F. The problem admits trivial ratio O(|T|^2). We consider the case |T|>k and prove that for directed undirected graphs and edge node-costs, a @r-approximation algorithm for Rooted Subsetk-Connectivity Augmentation implies the following approximation ratios for Subsetk-Connectivity Augmentation:(i)b(@r+k)+(|T||T|-k)^2O(log|T||T|-k), where b=1 for undirected graphs and b=2 for directed graphs. (ii)@r@?O(|T||T|-klogk). The best known values of @r on undirected graphs are min |T|,O(k) for edge-costs and min |T|,O(klog|T|) for node-costs; for directed graphs @r=|T| for both versions. Our results imply that unless k=|T|-o(|T|), Subsetk-Connectivity Augmentation admits the same ratios as the best known ones for the rooted version. This improves the ratios in Nutov (2009) [19] and Laekhanukit (2011) [15]. 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. 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. 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. We prove that, for every positive integer k, there is an integer N such that every 4-connected non-planar graph with at least N vertices has a minor isomorphic to K"4","k, the graph obtained from a cycle of length 2k+1 by adding an edge joining every pair of vertices at distance exactly k, or the graph obtained from a cycle of length k by adding two vertices adjacent to each other and to every vertex on the cycle. We also prove a version of this for subdivisions rather than minors, and relax the connectivity to allow 3-cuts with one side planar and of bounded size. We deduce that for every integer k there are only finitely many 3-connected 2-crossing-critical graphs with no subdivision isomorphic to the graph obtained from a cycle of length 2k by joining all pairs of diagonally opposite vertices. We further develop the study of testing graph properties as initiated by Goldreich, Goldwasser and Ron. Loosely speaking, given an oracle access to a graph, we wish to distinguish the case when the graph has a pre-determined property from the case when it is far from having this property. Whereas they view graphs as represented by their adjacency matrix and measure the distance between graphs as a fraction of all possible vertex pairs, we view graphs as represented by bounded-length incidence lists and measure the distance between graphs as a fraction of the maximum possible number of edges. Thus, while the previous model is most appropriate for the study of dense graphs, our model is most appropriate for the study of bounded-degree graphs. In particular, we present randomized algorithms for testing whether an unknown bounded-degree graph is connected, k-connected (for k > 1), cycle-free and Eulerian. Our algorithms work in time polynomial in 1 e, always accept the graph when it has the tested property, and reject with high probability if the graph is s-far from having the property. For example, the 2-connectivity algorithm rejects (with high probability) any N-vertex d-degree graph for which more than edN edges need to be added in order to make the graph 2-edge-connected. In addition we prove lower bounds of Ω(√N) on the query complexity of testing algorithms for the bipartite and expander properties. The visibility graph of a finite set of points in the plane has the points as vertices and an edge between two vertices if the line segment between them contains no other points. This paper establishes bounds on the edge- and vertex-connectivity of visibility graphs. Unless all its vertices are collinear, a visibility graph has diameter at most 2, and so it follows by a result of Plesnik (Acta Fac. Rerum Nat. Univ. Comen. Math. 30:71–93, 1975) that its edge-connectivity equals its minimum degree. We strengthen the result of Plesnik by showing that for any two vertices v and w in a graph of diameter 2, if deg(v)≤deg(w) then there exist deg(v) edge-disjoint vw-paths of length at most 4. For vertex-connectivity, we prove that every visibility graph with n vertices and at most l collinear vertices has connectivity at least @math , which is tight. We also prove the qualitatively stronger result that the vertex-connectivity is at least half the minimum degree. Finally, in the case that l=4 we improve this bound to two thirds of the minimum degree. A spanning tree in a graph is the smallest connected spanning subgraph. Given a graph, how does one find the smallest (i.e., least number of edges) 2-connected spanning subgraph (connectivity refers to both edge and vertex connectivity, if not specified)? Unfortunately, the problem is known to be NP-hard. We consider the problem of finding a better approximation to the smallest 2-connected subgraph, by an efficient algorithm. For 2-edge connectivity, our algorithm guarantees a solution that is no more than 3 2 times the optimal. For 2-vertex connectivity, our algorithm guarantees a solution that is no more than 5 3 times the optimal. The previous best approximation factor is 2 for each of these problems. The new algorithms (and their analyses) depend upon a structure called a carving of a graph, which is of independent interest. We show that approximating the optimal solution to within an additive constant is NP-hard as well. We also consider the case where the graph has edge weights. For this case, we show that an approximation factor of 2 is possible in polynomial time for finding a k -edge connected spanning subgraph. This improves an approximation factor of 3 for k = 2, due to Frederickson and Ja´Ja´ [1981], and extends it for any k (with an increased running time though). 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. An efficient heuristic is presented for the problem of finding a minimum-size k-connected spanning subgraph of an (undirected or directed) simple graph G=(V,E). There are four versions of the problem, and the approximation guarantees are as follows: minimum-size k-node connected spanning subgraph of an undirected graph 1 + [1 k], minimum-size k-node connected spanning subgraph of a directed graph 1 + [1 k], minimum-size k-edge connected spanning subgraph of an undirected graph 1+[2 (k+1)], minimum-size k-edge connected spanning subgraph of a directed graph 1 + [4 k ]. The heuristic is based on a subroutine for the degree-constrained subgraph (b-matching) problem. It is simple and deterministic and runs in time O(k|E|2). The following result on simple undirected graphs is used in the analysis: The number of edges required for augmenting a graph of minimum degree k to be k-edge connected is at most k,|V| (k+1). For undirected graphs and k=2, a (deterministic) parallel NC version of the heuristic finds a 2-node connected (or 2-edge connected) spanning subgraph whose size is within a factor of ( @math ) of minimum, where @math is a constant. 1. Fundamental Concepts. What Is a Graph? Paths, Cycles, and Trails. Vertex Degrees and Counting. Directed Graphs. 2. Trees and Distance. Basic Properties. Spanning Trees and Enumeration. Optimization and Trees. 3. Matchings and Factors. Matchings and Covers. Algorithms and Applications. Matchings in General Graphs. 4. Connectivity and Paths. Cuts and Connectivity. k-connected Graphs. Network Flow Problems. 5. Coloring of Graphs. Vertex Colorings and Upper Bounds. Structure of k-chromatic Graphs. Enumerative Aspects. 6. Planar Graphs. Embeddings and Euler's Formula. Characterization of Planar Graphs. Parameters of Planarity. 7. Edges and Cycles. Line Graphs and Edge-Coloring. Hamiltonian Cycles. Planarity, Coloring, and Cycles. 8. Additional Topics (Optional). Perfect Graphs. Matroids. Ramsey Theory. More Extremal Problems. Random Graphs. Eigenvalues of Graphs. Appendix A: Mathematical Background. Appendix B: Optimization and Complexity. Appendix C: Hints for Selected Exercises. Appendix D: Glossary of Terms. Appendix E: Supplemental Reading. Appendix F: References. Indices.