Abstract: Let F be a finite set of graphs. In the F-deletion problem, we are given an n-vertex graph G and an integer k as input, and asked whether at most k vertices can be deleted from G such that the resulting graph does not contain a graph from F as a minor. F-deletion is a generic problem and by selecting different sets of forbidden minors F, one can obtain various fundamental problems such as {\sc Vertex Cover}, {\sc Feedback Vertex Set} or {\sc Tree width k-deletion}. In this paper we obtain a number of generic algorithmic results about F-deletion, when F contains at least one planar graph. The highlights of our work are: 1. A constant factor approximation algorithm for the optimization version of Planar F-deletion, 2. A linear time and single exponential parameterized algorithm, that is, an algorithm running in time O(exp(2, O(k))n), for the parameterized version of Planar F-deletion where all graphs in Planar F are connected, 3. A polynomial kernel for parameterized F-deletion. These algorithms unify, generalize, and improve a multitude of results in the literature. Our main results have several direct applications, but also the methods we develop on the way have applicability beyond the scope of this paper. Our results -- constant factor approximation, polynomial kernelization and FPT algorithms -- are stringed together by a common theme of polynomial time preprocessing.

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 (J. Comb. Theory, Ser. B 63(1):65–110, 1995; J. Comb. Theory, Ser. B 92(2):325–357, 2004), there is a cubic 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: Graphs: Definitions and Concepts Fundamental Topology Matrix Representation and Computer Storage of Graphs Trees and Planar Graphs Algorithms of Graphs Directed Graphs Networks: Networks Complex Networks and Network Analysis Ecological Network Analysis-Advances Ecological Network Analysis-Methodology Agent-based Modeling: Agent-based Modeling Cell Automata Modeling of Pest Percolation Agent-based Modeling of Biological Community Agent-based Modeling of Ecological Problems.

Abstract: Simultaneous embedding is concerned with simultaneously representing a series of graphs sharing some or all vertices. This forms the basis for the visualization of dynamic graphs and thus is an important field of research. Recently there has been a great deal of work investigating simultaneous embedding problems both from a theoretical and a practical point of view. We survey recent work on this topic.

Abstract: We present new and improved data structures that answer exact node-to-node distance queries in planar graphs. Such data structures are also known as distance oracles. For any directed planar graph on n nodes with non-negative lengths we obtain the following:• Given a desired space allocation S e [n lg lg n, n2], we show how to construct in O(S) time a data structure of size O(S) that answers distance queries in O(n/√ S) time per query. The best distance oracles for planar graphs until the current work are due to Cabello (SODA 2006), Chen and Xu (STOC 2000), Djidjev (WG 1996), and Fakcharoenphol and Rao (FOCS 2001). For σ e (1, 4/3) and space S = nσ, we essentially improve the query time from n2/S to √n2/S.• As a consequence, we obtain an improvement over the fastest algorithm for k-many distances in planar graphs whenever k e [√n, n).• We provide a linear-space exact distance oracle for planar graphs with query time O(n1/2+e) for any constant e > 0. This is the first such data structure with provable sublinear query time.• For edge lengths ≥ 1, we provide an exact distance oracle of space O(n) such that for any pair of nodes at distance l the query time is O(min{l, √ n}). Comparable query performance had been observed experimentally but could not be explained theoretically.Our data structures with superlinear space are based on the following new tool: given a non-self-crossing cycle C with c = O(√n) nodes, we can preprocess G in O(n) time to produce a data structure of size O(n lg lg c) that can answer the following queries in O(c) time: for a query node u, output the distance from u to all the nodes of C. This data structure builds on and provides an alternative for a related data structure of Klein (SODA 2005), which reports distances to the boundary of a face, rather than a cycle.

Abstract: We present an $\mathcal O( n^{3/2})$ algorithm for minimizing the number of bends in an orthogonal drawing of a plane graph. It has been posed as a long standing open problem at Graph Drawing 2003, whether the bound of $\mathcal O(n^{7/4}\sqrt{\log n})$ shown by Garg and Tamassia in 1996 could be improved. To answer this question, we show how to solve the uncapacitated min-cost flow problem on a planar bidirected graph with bounded costs and face sizes in $\mathcal O(n^{3/2})$ time.

Abstract: In this paper, we consider the problem of representing planar graphs by polygons whose sides touch. We show that at least six sides per polygon are necessary by constructing a class of planar graphs that cannot be represented by pentagons. We also show that the lower bound of six sides is matched by an upper bound of six sides with a linear-time algorithm for representing any planar graph by touching hexagons. Moreover, our algorithm produces convex polygons with edges having at most three slopes and with all vertices lying on an O(n)×O(n) grid.

Abstract: For graph G, let źbw(G) denote the branchwidth of G and źgm(G) the largest integer g such that G contains a g×g grid as a minor. We show that źbw(G)≤3źgm(G) for every planar graph G. This is an improvement over the bound źbw(G)≤4źgm(G) due to Robertson, Seymour and Thomas. Our proof is constructive and implies quadratic time constant-factor approximation algorithms for planar graphs for both problems of finding a largest grid minor and of finding an optimal branch-decomposition: (3+∈)-approximation for the former and (2+∈)-approximation for the latter, where ∈ is an arbitrary positive constant. We also study the tightness of the above bound. We show that for any constant c<2, the bound of ${\mathop {\mathrm {bw}}}(G)\leq c\; {\mathop {\mathrm {gm}}}(G) + o({\mathop {\mathrm {gm}}}(G))$ does not hold in general for a planar graph G.

Abstract: Initial steps in the study of inner expansion properties of infinite Cayley graphs and other infinite graphs, such as hyperbolic ones, are taken, in a flavor similar to the well-known Lipton-Tarjan square root separation result for planar graphs. Connections to relaxed versions of quasi-isometries are explored, such as regular and semiregular maps.

Abstract: We describe the first algorithm to compute maximum flows in surface-embedded graphs in near-linear time. Specifically, given a graph embedded on a surface of genus $g$, with two specified vertices $s$ and $t$ and integer edge capacities that sum to $C$, our algorithm computes a maximum $(s,t)$-flow in $O(g^8 n\log^2 n\log^2 C)$ time. We also present a combinatorial algorithm that takes $g^{O(g)} n^{3/2}$ arithmetic operations. Except for the special case of planar graphs, for which an $O(n\log n)$-time algorithm has been known for 20 years, the best previous time bounds for maximum flows in surface-embedded graphs follow from algorithms for general sparse graphs. For graphs of any fixed genus, our algorithms improve these time bounds by roughly a factor of $\sqrt{n}$. Our key insight is to optimize the homology class of the flow, rather than directly optimizing the flow itself; two flows are in the same homology class if their difference is a weighted sum of directed facial cycles. A dual formulation of o...

Abstract: A conjecture of Carsten Thomassen states that every 4-connected line graph is hamiltonian. It is known that the conjecture is true for 7-connected line graphs. We improve this by showing that any 5-connected line graph of minimum degree at least 6 is hamiltonian. The result extends to claw-free graphs and to Hamilton-connectedness.

Abstract: We describe a data structure for submatrix maximum queries in Monge matrices or Monge partial matrices, where a query specifies a contiguous submatrix of the given matrix, and its output is the maximum element of that submatrix. Our data structure for an n x n Monge matrix takes O(n log n) space, O(n log2n) preprocessing time, and can answer queries in O(log2n) time. For a Monge partial matrix the space bound and the preprocessing time both grow by the small factor α(n), where α(n) is the inverse Ackermann function. Our design exploits an interpretation of the column maxima in a Monge matrix (resp., Monge partial matrix) as an upper envelope of pseudo-lines (resp., pseudo-segments).We give two applications for this data structure: (1) For a set of n points in a rectangle B in the plane, we build a data structure that, given a query point p, returns the largest-area empty axis-parallel rectangle contained in B and containing p, in O(log4n) time. The preprocessing time is O(nα(n) log4n), and the space required is O(nα(n) log3n). This improves substantially a previous data structure of Augustine et al. [arXiv: 1004.0558] that requires quadratic space. (2) Given an n-node arbitrarily weighted planar digraph, with possibly negative edge weights, we build, in O(n log2n/log log n) time, a linear-size data structure that supports edge-weight updates and distance queries between arbitrary pairs of nodes (where the distance is minimum weight of a path in the graph between the pair of nodes), in O(n2/3 log5/3n) time for each update and query. This improves the O(n4/5 log13/5n)-time bound of Fakcharoenphol and Rao [JCSS 72, 2006]. Our data structure has already been applied in a recent maximum flow algorithm for planar graphs of Borradaile et al. [FOCS 2011], and we believe it will find additional applications.

Abstract: We consider two operations on an edge of an embedded graph (or equivalently a ribbon graph): giving a half-twist to the edge and taking the partial dual with respect to the edge. These two operations give rise to an action of S3 jE(G)j , the ribbon group, on G. The action of the ribbon group on embedded graphs extends the concepts of duality, partial duality and Petrie duality. We show that this ribbon group action gives a complete characterization of duality in that if G is any cellularly embedded graph with medial graph Gm, then the orbit of G under the group action is precisely the set of all graphs with medial graphs isomorphic (as abstract graphs) to Gm. We provide characterizations of special sets of twisted duals, such as the partial duals, of embedded graphs in terms of medial graphs and we show how dierent kinds of graph isomorphism give rise to these various notions of duality. The ribbon group action then leads to a deeper understanding of the properties of, and relationships among, various graph polynomials via the generalized transition polynomial which interacts naturally with the ribbon group action. ribbon group action on the set of embedded graphs by taking duals and twisting individual edges. The ribbon group action completes the classical concept of duality for embedded graphs, and we exploit this group action to unravel and explore connections between embedded graphs and their medial graphs, and to determine implications of these new relations. The ribbon group action allows us to: classify the twisted duals of an embedded graph; characterize all graphs with isomorphic medial graphs under various kinds of isomorphism; extend the concept of a Tait graph to all 4-regular graphs; and to determine new properties of polynomials of embedded graphs. There are many investigations into planar and abstract graphs that have not yet been fully extended to the context of graphs embedded in surfaces. Consider for example the classical relations among the medial graphs and the duals of plane graphs. Suppose that G is a plane graph, G is its dual, and Gm is its medial graph. The medial graph of G is exactly the medial graph of G, i.e. (G )m = Gm, where \=" denotes equality as plane graphs. In fact, the connection between geometric duals and medial graphs is a little stronger than this. The two graphs G and G are the only plane graphs that have Gm as their plane medial graphs, that is,

Abstract: Two kinds of approximation algorithms exist for the k-BALANCED PARTITIONING problem: those that are fast but compute unsatisfactory approximation ratios, and those that guarantee high quality ratios but are slow. In this paper we prove that this tradeoff between runtime and solution quality is unavoidable. For the problem a minimum number of edges in a graph need to be found that, when cut, partition the vertices into k equal-sized sets. We develop a general reduction which identifies some sufficient conditions on the considered graph class in order to prove the hardness of the problem. We focus on two combinatorially simple but very different classes, namely trees and solid grid graphs. The latter are finite connected subgraphs of the infinite two-dimensional grid without holes. We apply the reduction to show that for solid grid graphs it is NP-hard to approximate the optimum number of cut edges within any satisfactory ratio. We also consider solutions in which the sets may deviate from being equal-sized. Our reduction is applied to grids and trees to prove that no fully polynomial time algorithm exists that computes solutions in which the sets are arbitrarily close to equal-sized. This is true even if the number of edges cut is allowed to increase when the limit on the set sizes decreases. These are the first bicriteria inapproximability results for the k-BALANCED PARTITIONING problem.

Abstract: These are lecture notes for the Current Developments in Mathematics conference at Harvard, November, 2011. We discuss topological, probabilistic and combinatorial aspects of the Laplacian on a graph embedded on a surface. The three main goals are to discuss: (1) for "circular" planar networks, the characterization due to Colin de Verdi\`ere of Dirichlet-to-Neumann operator; (2) The connections with the random spanning tree model; and (3) the characteristic polynomial of the Laplacian on an annulus and torus.

Abstract: We introduce the following notion of compressing an undirected graph G with (nonnegative) edge-lengths and terminal vertices R⊆V(G). A distance-preserving minor is a minor G′ (of G) with possibly different edge-lengths, such that R⊆V(G′) and the shortest-path distance between every pair of terminals is exactly the same in G and in G′. We ask: what is the smallest f*(k) such that every graph G with k=|R| terminals admits a distance-preserving minor G′ with at most f*(k) vertices? Simple analysis shows that f*(k)≤O(k4). Our main result proves that f*(k)≥Ω(k2), significantly improving over the trivial f*(k)≥k. Our lower bound holds even for planar graphs G, in contrast to graphs G of constant treewidth, for which we prove that O(k) vertices suffice.

Abstract: 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.

Abstract: Political districting is an intractable problem with significant ramifications for political representation. Districts often are required to satisfy some legal constraints, but these typically are not very restrictive, allowing decision makers to influence the composition of these districts without violating relevant laws. For example, while districts must often comprise a single contiguous area, a vast collection of acceptable solutions i.e., sets of districts remains. Choosing the best set of districts from this collection can be treated as a planar graph partitioning problem. When districts must be contiguous, successfully solving this problem requires an efficient computational method for evaluating contiguity constraints; common methods for assessing contiguity can require significant computation as the problem size grows. This paper introduces the geo-graph, a new graph model that ameliorates the computational burdens associated with enforcing contiguity constraints in planar graph partitioning when each vertex corresponds to a particular region of the plane. Through planar graph duality, the geo-graph provides a scale-invariant method for enforcing contiguity constraints in local search. Furthermore, geo-graphs allow district holes which typically are considered undesirable to be rigorously and efficiently integrated into the partitioning process.

Abstract: We give an O(n log3n) approximation scheme for Steiner forest in planar graphs, improving on the previous approximation scheme for this problem, which runs in O(nf(e)) time.

Abstract: We prove a complexity dichotomy theorem for symmetric complex-weighted Boolean #CSP when the constraint graph of the input must be planar. The problems that are #P-hard over general graphs but tractable over planar graphs are precisely those with a holographic reduction to matchgates. This generalizes a theorem of Cai, Lu, and Xia for the case of real weights. We also obtain a dichotomy theorem for a symmetric arity 4 signature with complex weights in the planar Holant framework, which we use in the proof of our #CSP dichotomy. In particular, we reduce the problem of evaluating the Tutte polynomial of a planar graph at the point (3,3) to counting the number of Eulerian orientations over planar 4-regular graphs to show the latter is #P-hard. This strengthens a theorem by Huang and Lu to the planar setting. Our proof techniques combine new ideas with refinements and extensions of existing techniques. These include planar pairings, the recursive unary construction, the anti-gadget technique, and pinning in the Hadamard basis.

Abstract: A well-known conjecture of Lovasz and Plummer from the mid-1970’s, still open, asserts that for every cubic graph G with no cutedge, the number of perfect matchings in G is exponential in |V (G)|. In this paper we prove the conjecture for planar graphs; we prove that if G is a planar cubic graph with no cutedge, then G has at least $$2^{{{\left| {V(G)} \right|} \mathord{\left/ {\vphantom {{\left| {V(G)} \right|} {655978752}}} \right. \kern- ulldelimiterspace} {655978752}}}$$ perfect matchings.

Abstract: 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.

Abstract: We show how to compute in O(n 4/3log 1/3 n+n 2/3 k 2/3log n) time the distance between k given pairs of vertices of a planar graph G with n vertices. This improves previous results whenever (n/log n)5/6≤k≤n 2/log 6 n. As an application, we speed up previous algorithms for computing the dilation of geometric planar graphs.

Abstract: For an integer k ≥ 0, suppose that each vertex v of a graph G has a set C(v) ⊆ {0,1, …, k} of labels, called a list of v. A list L(2,1)-labeling of G is an assignment of a label in C(v) to each vertex v of G such that every two adjacent vertices receive labels which differ by at least 2 and every two vertices of distance two receive labels which differ by at least 1. In this paper, we study the problem of reconfiguring one list L(2,1)-labeling of a graph into another list L(2,1)-labeling of the same graph by changing only one label assignment at a time, while at all times maintaining a list L(2,1)-labeling. First we show that this decision problem is PSPACE-complete, even for bipartite planar graphs and k ≥ 6. In contrast, we then show that the problem can be solved in linear time for general graphs if k ≤ 4. We finally consider the problem restricted to trees, and give a sufficient condition for which any two list L(2,1) -labelings of a tree can be transformed into each other.