Abstract: We develop the algorithmic theory of vertex separators and its relation to the embeddings of certain metric spaces. Unlike in the edge case, we show that embeddings into $L_1$ (and even Euclidean embeddings) are insufficient but that the additional structure provided by many embedding theorems does suffice for our purposes. We obtain an $O(\sqrt{\log n})$ approximation for minimum ratio vertex cuts in general graphs, based on a new semidefinite relaxation of the problem, and a tight analysis of the integrality gap which is shown to be $\Theta(\sqrt{\log n})$. We also prove an optimal $O(\log k)$-approximate max-flow/min-vertex-cut theorem for arbitrary vertex-capacitated multicommodity flow instances on $k$ terminals. For uniform instances on any excluded-minor family of graphs, we improve this to $O(1)$, and this yields a constant-factor approximation for minimum ratio vertex cuts in such graphs. Previously, this was known only for planar graphs, and for general excluded-minor families the best known ratio was $O(\log n)$. These results have a number of applications. We exhibit an $O(\sqrt{\log n})$ pseudoapproximation for finding balanced vertex separators in general graphs. In fact, we achieve an approximation ratio of $O(\sqrt{\log {opt}})$, where ${opt}$ is the size of an optimal separator, improving over the previous best bound of $O(\log {opt})$. Likewise, we obtain improved approximation ratios for treewidth: in any graph of treewidth $k$, we show how to find a tree decomposition of width at most $O(k \sqrt{\log k})$, whereas previous algorithms yielded $O(k \log k)$. For graphs excluding a fixed graph as a minor (which includes, e.g., bounded genus graphs), we give a constant-factor approximation for the treewidth. This in turn can be used to obtain polynomial-time approximation schemes for several problems in such graphs.

Abstract: We introduce a new method for computing massless Feynman integrals analytically in parametric form. An analysis of the method yields a criterion for a primitive Feynman graph $G$ to evaluate to multiple zeta values. The criterion depends only on the topology of $G$, and can be checked algorithmically. As a corollary, we reprove the result, due to Bierenbaum and Weinzierl, that the massless 2-loop 2-point function is expressible in terms of multiple zeta values, and generalize this to the 3, 4, and 5-loop cases. We find that the coefficients in the Taylor expansion of planar graphs in this range evaluate to multiple zeta values, but the non-planar graphs with crossing number 1 may evaluate to multiple sums with $6^\mathrm{th}$ roots of unity. Our method fails for the five loop graphs with crossing number 2 obtained by breaking open the bipartite graph $K_{3,4}$ at one edge.

Abstract: We give deterministic distributed algorithms that given i¾?> 0 find in a planar graph G, (1±i¾?)-approximations of a maximum independent set, a maximum matching, and a minimum dominating set. The algorithms run in O(log*|G|) rounds. In addition, we prove that no faster deterministic approximation is possible and show that if randomization is allowed it is possible to beat the lower bound for deterministic algorithms.

Abstract: We study discrete complex analysis and potential theory on a large family of planar graphs, the so-called isoradial ones. Along with discrete analogues of several classical results, we prove uniform convergence of discrete harmonic measures, Green's functions and Poisson kernels to their continuous counterparts. Among other applications, the results can be used to establish universality of the critical Ising and other lattice models.

Abstract: Greedy Routing is a class of routing algorithms in which the packets are forwarded in a manner that reduces the distance to the destination at every step. In an attempt to provide theoretical guarantees for a class of greedy routing algorithms, Papadimitriou and Ratajczak [19] came up with the following conjecture: Any 3-connected planar graph can be drawn in the plane such that for every pair of vertices s and t a distance decreasing path can be found. A path s = v1, v2,…,vk=t in a drawing is said to be distance decreasing if ||vi - t|| vi-1 -t||, 2 ≤ i ≤ k where || … || denotes the Euclidean distance. We settle this conjecture in the affirmative for the case of triangulations. A partitioning of the edges of a triangulation G into 3 trees, called the realizer of G, was first developed by Walter Schnyder who also gave a drawing algorithm based on this. We generalize Schnyder's algorithm to obtain a whole class of drawings of any given triangulation G. We show, using the Knaster-Kuratowski-Mazurkiewicz Theorem, that some drawing of G belonging to this class is greedy.

Abstract: Capacitated versions of Vertex Cover and DominatingSet have been studied intensively in terms of polynomial timeapproximation algorithms. Although the problems Dominating Set andVertex Cover have been subjected to considerable scrutiny in theparameterized complexity world, this is not true for their capacitatedversions. Here we make an attempt to understand the behavior of theproblems Capacitated Dominating Set and Capacitated VertexCover from the perspective of parameterized complexity. The original, uncapacitated versions of these problems, VertexCover and Dominating Set, are known to be fixed parameter tractablewhen parameterized by a structure of the graph called the treewidth (tw).In this paper we show that the capacitated versions of these problemsbehave differently. Our results are: - Capacitated Dominating Set is W[1]-hard when parameterized bytreewidth. In fact, Capacitated Dominating Set is W[1]-hard whenparameterized by both treewidth and solution size k of the capacitateddominating set. - Capacitated Vertex Cover is W[1]-hard when parameterized bytreewidth. - Capacitated Vertex Cover can be solved in time 2O(tw log k)nO(1)where tw is the treewidth of the input graph and k is the solution size.As a corollary, we show that the weighted version of Capacitated VertexCover in general graphs can be solved in time 2O(k log k)nO(1).This improves the earlier algorithm of Guo et al. [15] running intime O(1.2k2 + n2). Capacitated Vertex Cover is, therefore, to ourknowledge the first known "subset problem" which has turned out tobe fixed parameter tractable when parameterized by solution size butW[1]-hard when parameterized by treewidth.

Abstract: This thesis covers three aspects in the field of graph analysis and drawing. Firstly, the depth-first-search–based algorithm for finding triconnected components in general biconnected graphs is presented. This linear-time algorithm was originally published by Hopcroft and Tarjan [17], and corrected by Mutzel and Gutwenger [13]. Since the original paper is hard to understand, the algorithm is presented with illustrations to ease getting the vital ideas. Also, the crucial proposition is stated and proven in a way which is closer to the actual proceeding of the algorithm. Secondly, a simple linear-time algorithm for triangulating a biconnected planar graph is presented. Finally, a vertex-weighted variant of the so-called “shift-method” algorithm by de Fraysseix, Pach and Pollack [11] is introduced. The shift method is a linear-time algorithm to produce a straightline drawing of triangulated graphs on a grid with an area bound quadratic in the number of vertices of the graph. The original algorithm is modified to draw vertices as diamond shapes with area according to vertex weights. It is proven that the modified algorithm still produces a straight-line grid drawing of the graph in linear time with an area bound quadratic in the sum of vertex weights, and that edges do not cross the drawings of other vertices’ representations. The algorithm is presented within a framework to draw a special class of clustered graphs. The algorithm for finding triconnected components is implemented in JAVA for the yFiles graph drawing library [27]. The vertex-weighted shift method is implemented in JAVA for the visual analysis tool GEOMI [1].

Abstract: We consider the following variant of the classical random graph process introduced by Erdos and Renyi. Starting with an empty graph on n vertices, choose the next edge uniformly at random among all edges not yet considered, but only insert it if the graph remains planar. We show that for all e > 0, with high probability, t(n2) edges have to be tested before the number of edges in the graph reaches (1 + e)n. At this point, the graph is connected with high probability and contains a linear number of induced copies of any fixed connected planar graph, the first property being in contrast and the second one in accordance with the uniform random planar graph model. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008

Abstract: Wang and Lih conjectured that for every g>=5, there exists a number M(g) such that the square of a planar graph G of girth at least g and maximum degree @D>=M(g) is (@D+1)-colorable. The conjecture is known to be true for g>=7 but false for g@?{5,6}. We show that the conjecture for g=6 is off by just one, i.e., the square of a planar graph G of girth at least six and sufficiently large maximum degree is (@D+2)-colorable.

Abstract: We give an algorithm that, for a fixed graph H and integer k, decides whether an n-vertex H-minor-free graph G contains a path of length k in 2O(√k). nO(1) steps. Our approach builds on a combination of Demaine-Hajiaghayi's bounds on the size of an excluded grid in such graphs with a novel combinatorial result on certain branch decompositions of H-minor-free graphs. This result is used to bound the number of ways vertex disjoint paths can be routed through the separators of such decompositions. The proof is based on several structural theorems from the Graph Minors series of Robertson and Seymour. With a slight modification, similar combinatorial and algorithmic results can be derived for many other problems. Our approach can be viewed as a general framework for obtaining time 2O(√k). nO(1) algorithms on H-minor-free graph classes.

Abstract: A graph $G$ is {\em $k$-choosable} if for every assignment of a set $S(v)$ of $k$ colors to every vertex $v$ of $G$, there is a proper coloring of $G$ that assigns to each vertex $v$ a color from $S(v)$. We consider the complexity of deciding whether a given graph is $k$-choosable for some constant $k$. In particular, it is shown that deciding whether a given planar graph is 4-choosable is NP-hard, and so is the problem of deciding whether a given planar triangle-free graph is 3-choosable. We also obtain simple constructions of a planar graph which is not 4-choosable and a planar triangle-free graph which is not 3-choosable.

Abstract: The central problem of the total-colorings is the total-coloring conjecture, which asserts that every graph of maximum degree $\Delta$ admits a $(\Delta+2)$-total-coloring. Similar to edge-colorings—with Vizing's edge-coloring conjecture—this bound can be decreased by 1 for plane graphs of higher maximum degree. More precisely, it is known that if $\Delta\ge10$, then every plane graph of maximum degree $\Delta$ is $(\Delta+1)$-totally-colorable. On the other hand, such a statement does not hold if $\Delta\le3$. We prove that every plane graph of maximum degree 9 can be 10-totally-colored.

Abstract: Whether local algorithms can compute constant approximations of NP-hard problems is of both practical and theoretical interest. So far, no algorithms achieving this goal are known, as either the approximation ratio or the running time exceed O(1), or the nodes are provided with non-trivial additional information. In this paper, we present the first distributed algorithm approximating a minimum dominating set on a planar graph within a constant factor in constant time. Moreover, the nodes do not need any additional information.

Abstract: Distance-regularity of a graph is in general not determined by the spectrum of the graph. The spectral excess theorem states that a connected regular graph is distance-regular if for every vertex, the number of vertices at extremal distance (the excess) equals some given expression in terms of the spectrum of the graph. This result was proved by Fiol and Garriga [From local adjacency polynomials to locally pseudo-distance-regular graphs, J. Combinatorial Th. B 71 (1997), 162-183] using a local approach. This approach has the advantage that more general results can be proven, but the disadvantage that it is quite technical. The aim of the current paper is to give a less technical proof by taking a global approach.

Abstract: The intersection graph of a collection C of sets is a graph on the vertex set C, in which C1,C2 ∈ C are joined by an edge if and only if C1 ∩ C2 ≠ O. Erdos conjectured that the chromatic number of triangle-free intersection graphs of n segments in the plane is bounded from above by a constant. Here we show that it is bounded by a polylogarithmic function of n, which is the first nontrivial bound for this problem. More generally, we prove that for any t and k, the chromatic number of every Kk-free intersection graph of n curves in the plane, every pair of which have at most t points in common, is at most (ct log n/log k)c log k, where c is an absolute constant and ct only depends on t. We establish analogous results for intersection graphs of convex sets, x-monotone curves, semialgebraic sets of constant description complexity, and sets that can be obtained as the union of a bounded number of sets homeomorphic to a disk.Using a mix of results on partially ordered sets and planar separators, for large k we improve the best known upper bound on the number of edges of a k-quasi-planar topological graph with n vertices, that is, a graph drawn in the plane with curvilinear edges, no k of which are pairwise crossing. As another application, we show that for every e > 0 and for every positive integer t, there exist δ > 0 and a positive integer n0 such that every topological graph with n ≥ n0 vertices, at least n1+e edges, and no pair of edges intersecting in more than t points, has at least nδ pairwise intersecting edges.

Abstract: 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.

Abstract: In the k-Apex problem the task is to find at most k vertices whose deletion makes the given graph planar. The graphs for which there exists a solution form a minor closed class of graphs, hence by the deep results of Robertson and Seymour, there is an O(n^3) time algorithm for every fixed value of k. However, the proof is extremely complicated and the constants hidden by the big-O notation are huge. Here we give a much simpler algorithm for this problem with quadratic running time, by iteratively reducing the input graph and then applying techniques for graphs of bounded treewidth.

Abstract: In the first part of this paper, we provide polynomial quantum algorithms for additive approximations of the Tutte polynomial, at any point in the Tutte plane, for any planar graph. This includes an additive approximation of the partition function of the Potts model for any weighted planer graph at any temperature, as well as approximations to many other combinatorial graph properties described by the (multivariate or not) Tutte polynomial. To achieve these algorithms, we generalize the Temperley Lieb algebra representations, used in [6], to apply for any graph (not necessarily coming from a braid). Moreover, our representations are non-unitary, as are all representations of the Temperley Lieb algebra not corresponding to Jones polynomial related parameters. It might seem at first sight that this makes it impossible to apply them by a quantum circuit. We show how to do this nevertheless. The approximation window size turns out to be inverse polynomial in |G| times the product of the norms of the operators we apply. Additive approximations are tricky; the range of the possible outcomes, might be smaller than the size of the approximation window, in which case the outcome is meaningless. Unfortunately, ruling out this possibility is difficult: If we want to argue that our a are meaningful, we have to provide an estimate of the scale of the problem, which is difficult here exactly because no efficient algorithm for the problem exists! In the second part of the paper we provide an indirect but very convincing proof that our approximation is meaningful for a large range of parameters, by showing that in those cases, the problems

Abstract: The graph coloring is a classic NP-complete problem. Presently there is no effective method to solve this problem. Here we propose a modified particle swarm optimization (PSO) algorithm in which a disturbance factor is added to a particle swarm optimizer for improving its performance. When the current global best solution cannot be updated in a certain time period that is longer than the disturbance factor, a certain number of particles will be chosen according to probability and their velocities will be reset to force the particle swarm to get rid of local minimizers. It is found that this operation is helpful to improve the performance of particle swarm. Classic planar graph coloring problem is resolved by using modified particle swarm optimization algorithm. Numerical simulation results show that the performance of the modified PSO is superior to that of the classical PSO.

Abstract: Let the random graph Rn be drawn uniformly at random from the set of all simple planar graphs on n labelled vertices. We see that with high probability the maximum degree of Rn is Θ(ln n). We consider also the maximum size of a face and the maximum increase in the number of components on deleting a vertex. These results extend to graphs embeddable on any fixed surface.

Abstract: For every surface S (orientable or non-orientable), we give a linear time algorithm to test the graph isomorphism of two graphs, one of which admits an embedding of face-width at least 3 into S. This improves a previously known algorithm whose time complexity is nO(g), where g is the genus of S. This is the first algorithm for which the degree of polynomial in the time complexity does not depend on g. The above result is based on two linear time algorithms, each of which solves a problem that is of independent interest. The first of these problems is the following one. Let S be a fixed surface. Given a graph G and an integer k ≥ 3, we want to find an embedding of G in S of face-width at least k, or conclude that such an embedding does not exist. It is known that this problem is NP-hard when the surface is not fixed. Moreover, if there is an embedding, the algorithm can give all embeddings of face-width at least k, up to Whitney equivalence. Here, the face-width of an embedded graph G is the minimum number of points of G in which some non-contractible closed curve in the surface intersects the graph. In the proof of the above algorithm, we give a simpler proof and a better bound for the theorem by Mohar and Robertson concerning the number of polyhedral embeddings of 3-connected graphs. The second ingredient is a linear time algorithm for map isomorphism and Whitney equivalence. This part generalizes the seminal result of Hopcroft and Wong that graph isomorphism can be decided in linear time for planar graphs.

Abstract: A fullerene graph is a planar cubic graph whose all faces are pentagonal and hexagonal. The structure of cyclic edge-cuts of fullerene graphs of sizes at most 6 is known. In the paper we study cyclic 7-edge connectivity of fullerene graphs, distinguishing between degenerate and non-degenerate cyclic edge-cuts, regarding the arrangement of the 12 pentagons. We prove that if there exists a non-degenerate cyclic 7-edge-cut in a fullerene graph, then the graph is a nanotube unless it is one of the two exceptions presented. We determined that there are 57 configurations of degenerate cyclic 7-edge-cuts, and we listed all of them.

Abstract: Eigenvectors to the second smallest eigenvalue of the Laplace matrix of a graph, also known as Fiedler vectors, are the basic ingredient in spectral graph partitioning heuristics. Maximizing this second smallest eigenvalue over all nonnegative edge weightings with bounded total weight yields the absolute algebraic connectivity introduced by Fiedler, who proved tight connections of this value to the connectivity of the graph. Our objective is to gain a better understanding of the connections between separators and the eigenspace of this eigenvalue by studying the dual semidefinite optimization problem to the absolute algebraic connectivity. By exploiting optimality conditions we show that this problem is equivalent to finding an embedding of the $n$ nodes of the graph in $n$-space so that their barycenter is the origin, the distance between adjacent nodes is bounded by one, and the nodes are spread as much as possible (the sum of the squared norms is maximized). For connected graphs we prove that, for any separator in the graph, at least one of the two separated node sets is embedded in the shadow (with the origin being the light source) of the convex hull of the separator. Furthermore, we show that there always exists an optimal embedding whose dimension is bounded by the tree width of the graph plus one.

Abstract: In this paper, we introduce Vertex-face contact representation (VFCR for short) for 2-connected plane multigraphs. We present a simple linear time algorithm for constructing a VFCR for 2-connected plane graphs. Our algorithm only uses an st-orientation for Gand its corresponding st-orientation for the dual graph of G. We also show that one kind of vertex-vertex contact representation (VVCR) for 2-connected bipartite planar graphs introduced by Fraysseix et al. [2,3] can be easily obtained by applying our algorithm. In general, our algorithm produces a more compact representation than their algorithm. Then we investigate st-orientations for 3-connected planar graphs. We prove that a 3-connected planar graph Gwith nvertices and ffaces, has an st-orientation with the length of its longest directed path $\leq \frac{2n}{3}+2\lceil\sqrt{n/3}\rceil+5$. This implies that such a graph Gadmits a VFCR in a grid with non-trivial size bound. This non-trivial size bound also applies to the vertex-vertex contact representation [2,3] for a large class of 2-connected bipartite planar graphs.