Abstract: The algebraic connectivity of a graph is the second smallest eigenvalue of the graph Laplacian, and is a measure of how well-connected the graph is. We study the problem of adding edges (from a set of candidate edges) to a graph so as to maximize its algebraic connectivity. This is a difficult combinatorial optimization, so we seek a heuristic for approximately solving the problem. The standard convex relaxation of the problem can be expressed as a semidefinite program (SDP); for modest sized problems, this yields a cheaply computable upper bound on the optimal value, as well as a heuristic for choosing the edges to be added. We describe a new greedy heuristic for the problem. The heuristic is based on the Fiedler vector, and therefore can be applied to very large graphs.

Abstract: Spin systems are a general way to describe local interactions between nodes in a graph. In statistical mechanics, spin systems are often used as a model for physical systems. In computer science, they comprise an important class of families of combinatorial objects, for which approximate counting and sampling algorithms remain an elusive goal. The Dobrushin condition states that every row sum of the "influence matrix" for a spin system is less than 1 - epsiv, where epsiv > 0. This criterion implies rapid convergence (O(n log n) mixing time) of the single-site (Glauber) dynamics for a spin system, as well as uniqueness of the Gibbs measure. The dual criterion that every column sum of the influence matrix is less than 1 - epsiv has also been shown to imply the same conclusions. We examine a common generalization of these conditions, namely that the maximum eigenvalue of the influence matrix is less than 1 epsiv. Our main result is that this criterion implies O(n log n) mixing time for the Glauber dynamics. As applications, we consider the Ising model, hard-core lattice gas model, and graph colorings, relating the mixing time of the Glauber dynamics to the maximum eigenvalue for the adjacency matrix of the graph. For the special case of planar graphs, this leads to improved bounds on mixing time with quite simple proofs

Abstract: We introduce a new approach to design parameterized algorithms on planar graphs which builds on the seminal results of Robertson and Seymour on graph minors. Graph minors provide a list of powerful theoretical results and tools. However, the widespread opinion in the graph algorithms community about this theory is that it is of mainly theoretical importance. In this paper we show how deep min-max and duality theorems from graph minors can be used to obtain exponential speed-up to many known practical algorithms for different domination problems. Our use of branch-width instead of the usual tree-width allows us to obtain much faster algorithms. By using this approach, we show that the k-dominating set problem on planar graphs can be solved in time O(215.13 \sqrt k + n3).

Abstract: Let e>0 be a constant. For any edge-weighted planar graph G and a subset S of nodes of G, there is a subgraph H of G of weight a constant times that of the minimum Steiner tree for S such that distances in H between nodes in S are at most 1+e times the corresponding distances in G. As a consequence, there is an O(n log n)-time approximation scheme for finding a TSP among a given subset of nodes of a planar graph. This is the first PTAS for the problem.

Abstract: We study straight-line drawings of planar graphs with few segments and few slopes. Optimal results are obtained for all trees. Tight bounds are obtained for outerplanar graphs, 2-trees, and planar 3-trees. We prove that every 3-connected plane graph on $n$ vertices has a plane drawing with at most ${5/2}n$ segments and at most $2n$ slopes. We prove that every cubic 3-connected plane graph has a plane drawing with three slopes (and three bends on the outerface). In a companion paper, drawings of non-planar graphs with few slopes are also considered.

Abstract: We develop a new randomized method, random separation, for solving fixed-cardinality optimization problems on graphs, i.e., problems concerning solutions with exactly a fixed number k of elements (e.g., k vertices V') that optimize solution values (e.g., the number of edges covered by V'). The key idea of the method is to partition the vertex set of a graph randomly into two disjoint sets to separate a solution from the rest of the graph into connected components, and then select appropriate components to form a solution. We can use universal sets to derandomize algorithms obtained from this method. This new method is versatile and powerful as it can be used to solve a wide range of fixed-cardinality optimization problems for degree-bounded graphs, graphs of bounded degeneracy (a large family of graphs that contains degree-bounded graphs, planar graphs, graphs of bounded tree-width, and nontrivial minor-closed families of graphs), and even general graphs.

Abstract: The concept of typed attributed graphs and graph transformation is most significant for modeling and meta modeling in software engineering and visual languages, but up to now there is no adequate theory for this important branch of graph transformation. In this article we give a new formalization of typed attributed graphs, which allows node and edge attribution. The first main result shows that the corresponding category is isomorphic to the category of algebras over a specific kind of attributed graph structure signature. This allows to prove the second main result showing that the category of typed attributed graphs is an instance of "adhesive HLR categories". This new concept combines adhesive categories introduced by Lack and Sobocinski with the well-known approach of high-level replacement (HLR) systems using a new simplified version of HLR conditions. As a consequence we obtain a rigorous approach to typed attributed graph transformation providing as fundamental results the Local Church-Rosser, Parallelism, Concurrency, Embedding and Extension Theorem and a Local Confluence Theorem known as Critical Pair Lemma in the literature.

Abstract: We give the first correct O(n log n) algorithm for finding a maximum st-flow in a directed planar graph. After a preprocessing step that consists in finding single-source shortest-path distances in the dual, the algorithm consists of repeatedly saturating the leftmost residual s-to-t path.

Abstract: It is known that a planar graph on n vertices has branch-width-tree-width bounded by $\alpha \sqrt {n}$. In many algorithmic applications, it is useful to have a small bound on the constant α. We give a proof of the best, so far, upper bound for the constant α. In particular, for the case of tree-width, α < 3.182 and for the case of branch-width, α < 2.122. Our proof is based on the planar separation theorem of Alon, Seymour, and Thomas and some min–max theorems of Robertson and Seymour from the graph minors series. We also discuss some algorithmic consequences of this result. © 2005 Wiley Periodicals, Inc. J Graph Theory The first author is supported by Norges forskningsrad projects 162731-V00 and 160778-V30. The second author is supported by the Spanish CICYT project TIN-2004-07925 (GRAMMARS).

Abstract: Several basic theorems about the chromatic number of graphs can be extended to results in which, in addition to the existence of a κ-coloring, it is also shown that all κ-colorings of the graph in question are Kempe equivalent. Here, it is also proved that for a planar graph with chromatic number less than κ, all κ-colorings are Kempe equivalent.

Abstract: A \emph{clique} is a set of pairwise adjacent vertices in a graph. We determine the maximum number of cliques in a graph for the following graph classes: (1) graphs with $n$ vertices and $m$ edges; (2) graphs with $n$ vertices, $m$ edges, and maximum degree $\Delta$; (3) $d$-degenerate graphs with $n$ vertices and $m$ edges; (4) planar graphs with $n$ vertices and $m$ edges; and (5) graphs with $n$ vertices and no $K_5$-minor or no $K_{3,3}$-minor. For example, the maximum number of cliques in a planar graph with $n$ vertices is $8(n-2)$.

Abstract: We give a novel general approach for solving NP-hard optimization problems that combines dynamic programming and fast matrix multiplication. The technique is based on reducing much of the computation involved to matrix multiplication. We show that our approach works faster than the usual dynamic programming solution for any vertex subset problem on graphs of bounded branchwidth. In particular, we obtain the fastest algorithms for Planar Independent Set of runtime O(22.52√n), for PLANAR DOMINATING SET of runtime exact O(23.99√n) and parameterized O(211.98√k)ċ nO(1), and for PLANAR HAMILTONIAN Cycle of runtime O(25.58√n). The exponent of the running time is depending heavily on the running time of the fastest matrix multiplication algorithm that is currently o(n2.376).

Abstract: We investigate the computational complexity of the following problem. Given a planar graph in which some vertices have already been placed in the plane, place the remaining vertices to form a plana...

Abstract: A bisection of a graph with $n$ vertices is a partition of its vertices into two sets, each of size $n/2$. The bisection cost is the number of edges connecting the two sets. The problem of finding a bisection of minimum cost is prototypical to graph partitioning problems, which arise in numerous contexts. This problem is NP-hard. We present an algorithm that finds a bisection whose cost is within a factor of $O(\log^{1.5} n)$ from the minimum. For graphs excluding any fixed graph as a minor (e.g., planar graphs) we obtain an improved approximation ratio of $O(\log n)$. The previously known approximation ratio for bisection was roughly $\sqrt{n}$.

Abstract: As advanced technologies in wafer manufacturing push patterning processes toward lower-k 1 subwavelength printing, lithography for mass production potentially suffers from decreased patterning fidelity. This results in generation of many hotspots , which are actual device patterns with relatively large CD and image errors with respect to on-wafer targets. Hotspots can be formed under a variety of conditions such as the original design being unfriendly to the RET that is applied, unanticipated pattern combinations in rule-based OPC, or inaccuracies in model-based OPC. When these hotspots fall on locations that are critical to the electrical performance of a device, device performance and parametric yield can be significantly degraded. Previous rule-based hotspot detection methods suffer from long runtimes for complicated patterns. Also, the model generation process that captures process variation within simulation-based approaches brings significant overheads in terms of validation, measurement and parameter calibration. In this paper, we first describe a novel detection algorithm for hotspots induced by lithographic uncertainty. Our goal is to rapidly detect all lithographic hotspots without significant accuracy degradation. In other words, we propose a filtering method: as long as there are no "false negatives", i.e., we successfully have a superset of actual hotspots, then our method can dramatically reduce the layout area for golden hotspot analysis. The first step of our hotspot detection algorithm is to build a layout graph which reflects pattern-related CD variation. Given a layout L , the layout graph G = ( V , E c union E p ) consists of nodes V , corner edges E c and proximity edges E p . A face in the layout graph includes several close features and the edges between them. Edge weight can be calculated from a traditional 2-D model or a lookup table. We then apply a three-level hotspot detection: (1) edge-level detection finds the hotspot caused by two close features or " L -shaped" features; (2) face-level detection finds the pattern-related hotspots which span several close features; and (3) merged-face-level detection finds hotspots with more complex patterns. To find the merged faces which capture the pattern-related hotspots, we propose to convert the layout into a planar graph G . We then construct its dual graph G D and sort the dual nodes according to their weights. We merge the sorted dual nodes (i.e., the faces in G ) that share a given feature, in sequence. We have tested our flow on several industry testcases. The experimental results show that our method is promising: for a 90nm metal layer with 17 hotspots detected by commercial optical rule check (ORC) tools, our method can detect all of them while the overall runtime improvement is more than 287X.

Abstract: Let \varphi(X) be a first-order formula in the language of graphs that has a free set variable X, and assume that X only occurs positively in \varphi(X). Then a natural minimisation problem associated with \varphi(X) is to find, in a given graph G, a vertex set S of minimum size such that G satisfies \varphi(S). Similarly, if X only occurs negatively in \varphi(X), then \varphi(X) defines a maximisation problem. Many well-known optimisation problems are first-order definable in this sense, for example, MINIMUM DOMINATING SET or MAXIMUM INDEPENDENT SET. We prove that for each class C of graphs with excluded minors, in particular for each class of planar graphs, the restriction of a first-order definable optimisation problem to the class C has a polynomial time approximation scheme. A crucial building block of the proof of this approximability result is a version of Gaifman’s locality theorem for formulas positive in a set variable. This result may be of independent interest.

Abstract: We study various properties of a random graph R n , drawn uniformly at random from the class \( \mathcal{A}_n \) of all simple graphs on n labelled vertices that satisfy some given property, such as being planar or having tree-width at most κ In particular, we show that if the class \( \mathcal{A} \) is’ small’ and ‘addable’, then the probability that R n is connected is bounded away from 0 and from 1 As well as connectivity we study the appearances of subgraphs, and thus also vertex degrees and the numbers of automorphisms We see further that if \( \mathcal{A} \) is’ smooth’ then we can make much more precise statements for example concerning connectivity

Abstract: We present a randomized algorithm for finding maximum matchings in planar graphs in timeO(n ω/2), whereω is the exponent of the best known matrix multiplication algorithm. Sinceω<2.38, this algorithm breaks through theO(n 1.5) barrier for the matching problem. This is the first result of this kind for general planar graphs. We also present an algorithm for generating perfect matchings in planar graphs uniformly at random usingO(n ω/2) arithmetic operations. Our algorithms are based on the Gaussian elimination approach to maximum matchings introduced in [16].

Abstract: We show that the Kauffman bracket $[L]$ of a checkerboard colorable virtual link $L$ is an evaluation of the Bollobas-Riordan polynomial $R_{G_L}$ of a ribbon graph associated with $L$. This result generalizes Thistlethwaite's celebrated theorem relating the Kauffman bracket with the Tutte polynomial of planar graphs.

Abstract: Point-based shape representation has received increased attention in recent years, mainly due to its simplicity. One of the most fundamental operations for point set processing is to find the neighbors of each point. Mesh structures and neighborhood graphs are commonly used for this purpose. However, though meshes are very popular in the field of computer graphics, neighbor relations encoded in a mesh are often distorted. Likewise, neighborhood graphs, such as the minimum spanning tree (MST), relative neighborhood graph (RNG), and Gabriel graph (GG), are also imperfect as they usually give too few neighbors for a given point. In this paper, we introduce a generalization of Gabriel graph, named elliptic Gabriel graph (EGG), which takes an elliptic influence region instead of the circular region in GG. In order to determine the appropriate aspect ratio of the elliptic influence region of EGG, this paper also presents the analysis between the aspect ratio of the elliptic influence region and the average valence of the resulting neighborhood. Analytic and empirical test results are included.

Abstract: We study the maximum edge-disjoint paths problem in undirected planar graphs: given a graph G and node pairs s1t1, s2t2, ..., sktk, the goal is to maximize the number of pairs that can be connected (routed) by edge-disjoint paths. The natural multicommodity flow relaxation has an Ω(√n) integrality gap. Motivated by this, we consider solutions with small constant congestion c > 1; that is, solutions in which up to c paths are allowed to use an edge (alternatively, each edge has a capacity of c). In previous work we obtained an O(log n) approximation with congestion 2 via the flow relaxation. This was based on a method of decomposing into well-linked subproblems.In this paper we obtain an O(1) approximation with congestion 4. To obtain this improvement we develop an alternative decomposition that is specific to planar graphs. The decomposition produces instances that we call Okamura-Seymour (OS) instances. These have the property that all terminals lie on a single face. Another ingredient we develop is a constant factor approximation for the all-or-nothing flow problem on OS instances via the flow relaxation.We also study limitations on the approximation that can be achieved by a well-linked decomposition. For general graphs we show a lower bound of Ω(log n). For planar graphs we describe instances that suggest a super-constant lower bound.

Abstract: A universal TSP tour of a metric space is a total ordering of the points of the space such that for any finite subset, the tour which visits these points in the given order is not too much longer than the optimal tour. There is a vast literature on the TSP problem, and universal TSP tours have been studied since the 1980's when Bartholdi and Platzman [29] introduced the spacefilling curve heuristic for the Euclidean TSP problem and conjectured that there exists a constant-competitive universal TSP tour based on this heuristic. Here, we settle this conjecture negatively by proving an Ω (6√log n/log log n) lower bound for universal TSP tours of the n × n grid; this is the first known example of a family of finite metrics with no constant-competitive universal tour.Generalizing from the n × n grid to arbitrary weighted planar graph metrics, and more generally H-minor-free metrics, we improve the best known upper bound for universal tours of such metrics from O(log4n/ log log n) to O(log2n).

Abstract: We study Kronrod-Reeb graphs of functions with isolated critical points on smooth manifolds. We prove that any finite graph, which satisfies the condition ℑ is a Kronrod-Reeb graph for some such function on some manifold. In this connection, monotone functions on graphs are investigated.