Common quality metrics of graph drawing have been about the readability criteria, such as small number of edge crossings, small drawing area and small total edge length. Bold graph drawing considers more realistic drawings consisting of vertices as disks of some radius and edges as rectangles of some width. However, the relationship that links these readability criteria with the rendering criteria in node-link diagrams has still not been well-established. This paper introduces a model, so-called INKA (Ink-Active), that encapsulates mathematically the relationship between all common drawing factors. Consequently, we investigate our INKA model on several common drawing algorithms and real-world graphs.
A number of graph drawing algorithms have aimed for a minimum total edge length, or more precisely, a minimum amount of ink. This criterion has been studied @cite_22 @cite_6 @cite_7 @cite_23 .
@cite_7 @cite_23 @cite_22 @cite_6
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This paper describes algorithms for computing non-planar drawings of planar graphs in subquadratic area such that: (i) edge crossings are allowed only if they create large angles; (ii) the maximum number of bends per edge is bounded by a (small) constant. A 5 4-approximation algorithm is presented for the minimum cardinality 2-edge-connected spanning subgraph problem in undirected graphs. This improves the previous best approximation ratio of 4 3. It is shown that our ratio is tight with respect to current lower bounds, and any further improvement is possible only if new lower bounds are discovered. We present an algorithm for finding the minimum cut of an undirected edge-weighted graph. It is simple in every respect. It has a short and compact description, is easy to implement, and has a surprisingly simple proof of correctness. Its runtime matches that of the fastest algorithm known. The runtime analysis is straightforward. In contrast to nearly all approaches so far, the algorithm uses no flow techniques. Roughly speaking, the algorithm consists of about | V | nearly identical phases each of which is a maximum adjacency search . We present the best known algorithms for approximating the minimum-size undirected @math -edge connected spanning subgraph. For simple graphs our approximation ratio is @math . The more precise version of this bound requires @math , and for all such @math it improves the long-standing performance ratio of Cheriyan and Thurimella [SIAM J. Comput., 30 (2000), pp. 528-560], @math . The improvement comes in two steps. First we show that for simple @math -edge connected graphs, any laminar family of degree @math sets is smaller than the general bound ( @math versus @math ). This immediately implies that iterated rounding improves the performance ratio of Cheriyan and Thurimella. The second step carefully chooses good edges for rounding. For multigraphs our approximation ratio is @math . This improves the previous ratio @math [H. N. Gabow, M. X. Goemans, E. Tardos, and D. P. Williamson, Networks, 53 (2009), pp. 345-357]. It is of interest since it is known that for some constant @math , an approximation ratio @math implies @math . Our approximation ratio extends to the minimum-size Steiner network problem, where @math denotes the average vertex demand. The algorithm exploits rounding properties of the first two linear programs in iterated rounding. Consider a drawing of a graph G in the plane such that crossing edges are coloured differently. The minimum number of colours, taken over all drawings of G, is the classical graph parameter thickness. By restricting the edges to be straight, we obtain the geometric thickness. By additionally restricting the vertices to be in convex position, we obtain the book thickness. This paper studies the relationship between these parameters and treewidth. Our first main result states that for graphs of treewidth k, the maximum thickness and the maximum geometric thickness both equal ⌈k 2⌉. This says that the lower bound for thickness can be matched by an upper bound, even in the more restrictive geometric setting. Our second main result states that for graphs of treewidth k, the maximum book thickness equals k if k ≤ 2 and equals k + 1 if k ≥ 3. This refutes a conjecture of Ganley and Heath [Discrete Appl. Math. 109(3):215-221, 2001]. Analogous results are proved for outerthickness, arboricity, and star-arboricity. We describe the first algorithms to compute minimum cuts in surface-embedded graphs in near-linear time. Given an undirected graph embedded on an orientable surface of genus g, with two specified vertices s and t, our algorithm computes a minimum (s,t)-cut in gO(g) n log n time. Except for the special case of planar graphs, for which O(n log n)-time algorithms have been known for more than 20 years, the best previous time bounds for finding minimum cuts in embedded graphs follow from algorithms for general sparse graphs. A slight generalization of our minimum-cut algorithm computes a minimum-cost subgraph in every Z2-homology class. We also prove that finding a minimum-cost subgraph homologous to a single input cycle is NP -hard. We study the problem of the amount of information required to draw a complete or a partial map of a graph with unlabeled nodes and arbitrarily labeled ports. A mobile agent, starting at any node of an unknown connected graph and walking in it, has to accomplish one of the following tasks: draw a complete map of the graph, i.e., find an isomorphic copy of it including port numbering, or draw a partial map, i.e., a spanning tree, again with port numbering. The agent executes a deterministic algorithm and cannot mark visited nodes in any way. None of these map drawing tasks is feasible without any additional information, unless the graph is a tree. Hence we investigate the minimum number of bits of information (minimum size of advice) that has to be given to the agent to complete these tasks. It turns out that this minimum size of advice depends on the number n of nodes or the number m of edges of the graph, and on a crucial parameter @m, called the multiplicity of the graph, which measures the number of nodes that have an identical view of the graph. We give bounds on the minimum size of advice for both above tasks. For @m=1 our bounds are asymptotically tight for both tasks and show that the minimum size of advice is very small. For @m>1 the minimum size of advice increases abruptly. In this case our bounds are asymptotically tight for topology recognition and asymptotically almost tight for spanning tree construction. This paper present a new approach to finding minimum cuts in undirected graphs. The fundamental principle is simple: the edges in a graph's minimum cut form an extremely small fraction of the graph's edges. Using this idea, we give a randomized, strongly polynomial algorithm that finds the minimum cut in an arbitrarily weighted undirected graph with high probability. The algorithm runs in O(n 2 log 3 n) time, a significant improvement over the previous O˜(mn) time bounds based on maximum flows. It is simple and intuitive and uses no complex data structures. Our algorithm can be parallelized to run in RNC with n 2 processors; this gives the first proof that the minimum cut problem can be solved in RNC . The algorithm does more than find a single minimum cut; it finds all of them. With minor modifications, our algorithm solves two other problems of interest. Our algorithm finds all cuts with value within a multiplicative factor of α of the minimum cut's in expected O˜(n 2α ) time, or in RNC with n 2α processors. The problem of finding a minimum multiway cut of graph into r pieces is solved in expected O˜(n 2(r-1) ) time, or in RNC with n 2(r-1) processors. The “trace” of the algorithm's execution on these two problems forms a new compact data structure for representing all small cuts and all multiway cuts in a graph. This data structure can be efficiently transformed into the more standard cactus representing for minimum cuts. The minimum cut problem for an undirected edge-weighted graph asks us to divide its set of nodes into two blocks while minimizing the weight sum of the cut edges. Here, we introduce a linear-time algorithm to compute near-minimum cuts. Our algorithm is based on cluster contraction using label propagation and Padberg and Rinaldi’s contraction heuristics [SIAM Review, 1991]. We give both sequential and shared-memory parallel implementations of our algorithm. Extensive experiments on both real-world and generated instances show that our algorithm finds the optimal cut on nearly all instances significantly faster than other state-of-the-art exact algorithms, and our error rate is lower than that of other heuristic algorithms. In addition, our parallel algorithm runs a factor 7.5× faster on average when using 32 threads. To further speed up computations, we also give a version of our algorithm that performs random edge contractions as preprocessing. This version achieves a lower running time and better parallel scalability at the expense of a higher error rate. We significantly improve known time bounds for solving the minimum cut problem on undirected graphs. We use a "semiduality" between minimum cuts and maximum spanning tree packings combined with our previously developed random sampling techniques. We give a randomized (Monte Carlo) algorithm that finds a minimum cut in an m -edge, n -vertex graph with high probability in O (m log 3 n ) time. We also give a simpler randomized algorithm that finds all minimum cuts with high probability in O( m log 3 n ) time. This variant has an optimal RNC parallelization. Both variants improve on the previous best time bound of O ( n 2 log 3 n ). Other applications of the tree-packing approach are new, nearly tight bounds on the number of near-minimum cuts a graph may have and a new data structure for representing them in a space-efficient manner. An exact formula is given for the maximum number of edges in a graph that admits a three-dimensional grid-drawing contained in a given bounding box. The first universal lower bound on the volume of threedimensional grid-drawings is obtained as a corollary. Our results generalise to the setting of multi-dimensional polyline grid-drawings. We present a deterministic algorithm that computes the edge-connectivity of a graph in near-linear time. This is for a simple undirected unweighted graph G with n vertices and m edges. This is the first o(mn) time deterministic algorithm for the problem. Our algorithm is easily extended to find a concrete minimum edge-cut. In fact, we can construct the classic cactus representation of all minimum cuts in near-linear time. The previous fastest deterministic algorithm by Gabow from STOC '91 took O(m+λ2 n), where λ is the edge connectivity, but λ can be as big as n−1. Karger presented a randomized near-linear time Monte Carlo algorithm for the minimum cut problem at STOC’96, but the returned cut is only minimum with high probability. Our main technical contribution is a near-linear time algorithm that contracts vertex sets of a simple input graph G with minimum degree Δ, producing a multigraph Ḡ with O(m Δ) edges, which preserves all minimum cuts of G with at least two vertices on each side. In our deterministic near-linear time algorithm, we will decompose the problem via low-conductance cuts found using PageRank a la Brin and Page (1998), as analyzed by Andersson, Chung, and Lang at FOCS’06. Normally, such algorithms for low-conductance cuts are randomized Monte Carlo algorithms, because they rely on guessing a good start vertex. However, in our case, we have so much structure that no guessing is needed. We give a ( 17 12 ) -approximation algorithm for the following NP- hard problem: Given a simple undirected graph, find a 2-edge connected span- ning subgraph that has the minimum number of edges. The best previous approximation guarantee was ( 3 2 ) . If the well known TSP ( 4 3 ) conjecture holds, then there is a ( 4 3 ) -approximation algorithm. Thus our main result gets half-way to this target. The minimum 3-way cut problem in an edge-weighted hypergraph is to find a partition of the vertices into 3 nonempty sets minimizing the total weight of hyperedges that have at least two endpoints in two different sets. In this paper we show that a minimum 3-way cut in hypergraphs can be found by using O(n^3) hypergraph minimum (s,t) cut computations, where n is the number of vertices in the hypergraph. Our simple algorithm is the first polynomial-time algorithm for finding minimum 3-way cuts in hypergraphs. We give improved approximations for two classical embedding problems: (i) minimizing the number of crossings in a drawing on the plane of a bounded degree graph; and (ii) minimizing the VLSI layout area of a graph of maximum degree four. These improved algorithms can be applied to improve a variety of VLSI layout problems. Our results are as follows. (i) We compute a drawing on the plane of a bounded degree graph in which the sum of the numbers of vertices and crossings is O(log3 n) @math times the minimum layout area. This is an O(log2 n) improvement over the best known long-standing result. 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. Let fd (G) denote the minimum number of edges that have to be added to a graph G to transform it into a graph of diameter at most d. We prove that for any graph G with maximum degree D and n > n0 (D) vertices, f2(G) = n - D - 1 and f3(G) ≥ n - O(D3). For d ≥ 4, fd (G) depends strongly on the actual structure of G, not only on the maximum degree of G. We prove that the maximum of fd (G) over all connected graphs on n vertices is n-⌊d-2 ⌋ - O(1). As a byproduct, we show that for the n-cycle Cn, fd (Cn) = n-(2⌊d-2 ⌋ - 1) - O(1) for every d and n, improving earlier estimates of Chung and Garey in certain ranges. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 161–172, 2000 1991 Mathematics Subject classification: 05C12. The best approximation algorithm for Max Cut in graphs of maximum degree 3 uses semidefinite programming, has approximation ratio 0.9326, and its running time is @Q(n^3^.^5logn); but the best combinatorial algorithms have approximation ratio 4 5 only, achieved in O(n^2) time [J.A. Bondy, S.C. Locke, J. Graph Theory 10 (1986) 477-504; E. Halperin, et al, J. Algorithms 53 (2004) 169-185]. Here we present an improved combinatorial approximation, which is a 5 6-approximation algorithm that runs in O(n^2) time, perhaps improvable even to O(n). Our main tool is a new type of vertex decomposition for graphs of maximum degree 3. We give simple, deterministic, distributed algorithms for computing maximal matchings, maximal independent sets and colourings. We show that edge colourings with at most 2Δ-1 colours, and maximal matchings can be computed within O(log* n + Δ) deterministic rounds, where Δ is the maximum degree of the network. We also show how to find maximal independent sets and (Δ + 1)-vertex colourings within O(log* n + Δ2) deterministic rounds. All hidden constants are very small and the algorithms are very simple. Partial Edge Drawing (PED) is a popular graph drawing style aimed at reducing edge crossings and visual clutter. PEDs are straight-line drawings where the central part of each edge is erased, and the length of the two remaining segments are computed so as to preserve useful geometric information. Recent studies on this approach focus on symmetric and δ-homogeneous PEDs (δ-SHPEDs). Given a straight-line drawing, a δ-SHPED of this drawing is immediately defined for a fixed value of δ. In particular, some edge crossings may not be avoidable, although the amount of ink removed from the original drawing might be large (e.g., 50 when δ = 1 4). On the other hand, it is possible to maximize the ink and remove edge crossings by renouncing to homogeneity. We present heuristics to produce symmetric PEDs that are either crossing-free or where crossings forming large angles are allowed. We also describe a user study in which PEDs obtained via our heuristics are compared with the standard model 1 4-SHPED. Our results suggest that the benefit of homogeneity overcomes in terms of readability the benefit of fewer crossings and more ink.