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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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 . 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. 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. 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. 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. 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. A dominating set in a graph G is a connected dominating set of G if it induces a connected subgraph of G. The connected domatic number of G is the maximum number of pairwise disjoint, connected dominating sets in V(G). We establish a sharp lower bound on the number of edges in a connected graph with a given order and given connected domatic number. We also show that a planar graph has connected domatic number at most 4 and give a characterization of planar graphs having connected domatic number 3. A Right Angle Crossing Graph (also called RAC graph for short) is a graph that has a straight-line drawing where any two crossing edges are orthogonal to each other. A 1-planar graph is a graph that has a drawing where every edge is crossed at most once. We study the relationship between RAC graphs and 1-planar graphs in the extremal case that the RAC graphs have as many edges as possible. It is known that a maximally dense RAC graph with n>3 vertices has 4n --- 10 edges. We show that every maximally dense RAC graph is 1-planar. Also, we show that for every integer i such that i≥0, there exists a 1-planar graph with n=8+4i vertices and 4n --- 10 edges that is not a RAC graph. 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. 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. Untangling is a process in which some vertices in a drawing of a planar graph are moved to obtain a straight-line plane drawing. The aim is to move as few vertices as possible. We present an algorithm that untangles the cycle graph C n while keeping Ω(n 2 3) vertices fixed. For any connected graph G, we also present an upper bound on the number of fixed vertices in the worst case. The bound is a function of the number of vertices, maximum degree, and diameter of G. One consequence is that every 3-connected planar graph has a drawing δ such that at most O((nlog n)2 3) vertices are fixed in every untangling of δ. 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. The bend-numberb(G) of a graph G is the minimum k such that G may be represented as the edge intersection graph of a set of grid paths with at most k bends. We confirm a conjecture of Biedl and Stern showing that the maximum bend-number of outerplanar graphs is 2. Moreover we improve the formerly known lower and upper bound for the maximum bend-number of planar graphs from 2 and 5 to 3 and 4, respectively. 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 study contact representations of edge-weighted planar graphs, where vertices are rectangles or rectilinear polygons and edges are polygon contacts whose lengths represent the edge weights. We show that for any given edge-weighted planar graph whose outer face is a quadrangle, that is internally triangulated and that has no separating triangles we can construct in linear time an edge-proportional rectangular dual if one exists and report failure otherwise. For a given combinatorial structure of the contact representation and edge weights interpreted as lower bounds on the contact lengths, a corresponding contact representation that minimizes the size of the enclosing rectangle can be found in linear time. If the combinatorial structure is not fixed, we prove NP-hardness of deciding whether a contact representation with bounded contact lengths exists. Finally, we give a complete characterization of the rectilinear polygon complexity required for representing biconnected internally triangulated graphs: For outerplanar graphs complexity 8 is sufficient and necessary, and for graphs with two adjacent or multiple non-adjacent internal vertices the complexity is unbounded. Abstract A drawing of a graph is 1-planar if each edge is crossed at most once. A graph is 1-planar if it has a 1-planar drawing. A k-bend RAC (Right Angle Crossing) drawing of a graph is a polyline drawing where each edge has at most k bends and edges cross only at right angles. A graph is k-bend RAC if it has a k -bend RAC drawing. A 0-bend RAC graph (drawing) is also called a straight-line RAC graph (drawing) . The relationships between 1-planar and k -bend RAC graphs have been partially studied in the literature. It is known that there are both 1-planar graphs that are not straight-line RAC and straight-line RAC graphs that are not 1-planar. The existence of 1-planar straight-line RAC drawings has been proven only for restricted families of 1-planar graphs. Two of the main questions still open are: ( i ) What is the complexity of deciding whether a graph has a drawing that is both 1-planar and straight-line RAC? ( i i ) Does every 1-planar graph have a drawing that is both 1-planar and 1-bend RAC? In this paper we answer these two questions. Namely, we prove an NP-hardness result for the first question, and we positively answer the second question by describing a drawing algorithm for 1-planar graphs. Given a graph G with n vertices and a set S of n points in the plane, a point-set embedding of G on S is a planar drawing such that each vertex of G is mapped to a distinct point of S. A geometric point-set embedding is a point-set embedding with no edge bends. This paper studies the following problem: The input is a set S of n points, a planar graph G with n vertices, and a geometric point-set embedding of a subgraph G^'@?G on a subset of S. The desired output is a point-set embedding of G on S that includes the given partial drawing of G^'. We concentrate on trees and show how to compute the output in O(n^2logn) time in a real-RAM model and with at most n-k edges with at most [email protected]?k [email protected]? bends, where k is the number of vertices of the given subdrawing. We also prove that there are instances of the problem which require at least k-3 bends on n-k edges. 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.