Abstract: In a parameterized problem, every instance I comes with a positive integer k. The problem is said to admit a polynomial kernel if, in polynomial time, one can reduce the size of the instance I to a polynomial in k while preserving the answer. In this work, we give two meta-theorems on kernelization. The first theorem says that all problems expressible in counting monadic second-order logic and satisfying a coverability property admit a polynomial kernel on graphs of bounded genus. Our second result is that all problems that have finite integer index and satisfy a weaker coverability property admit a linear kernel on graphs of bounded genus. These theorems unify and extend all previously known kernelization results for planar graph problems.

Abstract: We propose a network-filtering method, the Triangulated Maximally Filtered Graph (TMFG), that provides an approximate solution to the Weighted Maximal Planar Graph problem. The underlying idea of TMFG consists in building a triangulation that maximizes a score function associated with the amount of information retained by the network. TMFG uses as weights any arbitrary similarity measure to arrange data into a meaningful network structure that can be used for clustering, community detection and modeling. The method is fast, adaptable and scalable to very large datasets, it allows online updating and learning as new data can be inserted and deleted with combinations of local and non-local moves. TMFG permits readjustments of the network in consequence of changes in the strength of the similarity measure. The method is based on local topological moves and can therefore take advantage of parallel and GPUs computing. We discuss how this network-filtering method can be used intuitively and efficiently for big data studies and its significance from an information-theoretic perspective.

Abstract: Basic Methods: Seven Is More Than Six. The Pigeon-Hole Principle One Step at a Time. The Method of Mathematical Induction Enumerative Combinatorics: There Are a Lot of Them. Elementary Counting Problems No Matter How You Slice It. The Binomial Theorem and Related Identities Divide and Conquer. Partitions Not So Vicious Cycles. Cycles in Permutations You Shall Not Overcount. The Sieve A Function is Worth Many Numbers. Generating Functions Graph Theory: Dots and Lines. The Origins of Graph Theory Staying Connected. Trees Finding a Good Match. Coloring and Matching Do Not Cross. Planar Graphs Horizons: Does It Clique? Ramsey Theory So Hard to Avoid. Subsequence Conditions on Permutations Who Knows What It Looks Like, but It Exists. The Probabilistic Method At Least Some Order. Partial Orders and Lattices The Sooner The Better. Combinatorial Algorithms Does Many Mean More Than One? Computational Complexity.

Abstract: We give the first polynomial-time approximation schemes (PTASs) for the following problems: (1) uniform facility location in edge-weighted planar graphs; (2) $k$-median and $k$-means in edge-weighted planar graphs; (3) $k$-means in Euclidean spaces of bounded dimension. Our first and second results extend to minor-closed families of graphs. All our results extend to cost functions that are the $p$-th power of the shortest-path distance. The algorithm is local search where the local neighborhood of a solution $S$ consists of all solutions obtained from $S$ by removing and adding $1/\epsilon^{O(1)}$ centers.

Abstract: Basic Concepts and Algorithms Basic Concepts in Graph Theory and Algorithms Subramanian Arumugam and Krishnaiyan "KT" Thulasiraman Basic Graph Algorithms Krishnaiyan "KT" Thulasiraman Depth-First Search and Applications Krishnaiyan "KT" Thulasiraman Flows in Networks Maximum Flow Problem F. Zeynep Sargut, Ravindra K. Ahuja, James B. Orlin, and Thomas L. Magnanti Minimum Cost Flow Problem Balachandran Vaidyanathan, Ravindra K. Ahuja, James B. Orlin, and Thomas L. Magnanti Multi-Commodity Flows Balachandran Vaidyanathan, Ravindra K. Ahuja, James B. Orlin, and Thomas L. Magnanti Algebraic Graph Theory Graphs and Vector Spaces Krishnaiyan "KT" Thulasiraman and M.N.S. Swamy Incidence, Cut, and Circuit Matrices of a Graph Krishnaiyan "KT" Thulasiraman and M.N.S. Swamy Adjacency Matrix and Signal Flow Graphs Krishnaiyan "KT" Thulasiraman and M.N.S. Swamy Adjacency Spectrum and the Laplacian Spectrum of a Graph R. Balakrishnan Resistance Networks, Random Walks, and Network Theorems Krishnaiyan "KT" Thulasiraman and Mamta Yadav Structural Graph Theory Connectivity Subramanian Arumugam and Karam Ebadi Connectivity Algorithms Krishnaiyan "KT" Thulasiraman Graph Connectivity Augmentation Andras Frank and Tibor Jordan Matchings Michael D. Plummer Matching Algorithms Krishnaiyan "KT" Thulasiraman Stable Marriage Problem Shuichi Miyazaki Domination in Graphs Subramanian Arumugam and M. Sundarakannan Graph Colorings Subramanian Arumugam and K. Raja Chandrasekar Planar Graphs Planarity and Duality Krishnaiyan "KT" Thulasiraman and M.N.S. Swamy Edge Addition Planarity Testing Algorithm John M. Boyer Planarity Testing Based on PC-Trees Wen-Lian Hsu Graph Drawing Md. Saidur Rahman and Takao Nishizeki Interconnection Networks Introduction to Interconnection Networks S.A. Choudum, Lavanya Sivakumar, and V. Sunitha Cayley Graphs S. Lakshmivarahan, Lavanya Sivakumar, and S.K. Dhall Graph Embedding and Interconnection Networks S.A. Choudum, Lavanya Sivakumar, and V. Sunitha Special Graphs Program Graphs Krishnaiyan "KT" Thulasiraman Perfect Graphs Chinh T. Hoang and R. Sritharan Tree-Structured Graphs Andreas Brandstadt and Feodor F. Dragan Partitioning Graph and Hypergraph Partitioning Sachin B. Patkar and H. Narayanan Matroids Matroids H. Narayanan and Sachin B. Patkar Hybrid Analysis and Combinatorial Optimization H. Narayanan Probabilistic Methods, Random Graph Models, and Randomized Algorithms Probabilistic Arguments in Combinatorics C.R. Subramanian Random Models and Analyses for Chemical Graphs Daniel Pascua, Tina M. Kouri, and Dinesh P. Mehta Randomized Graph Algorithms: Techniques and Analysis Surender Baswana and Sandeep Sen Coping with NP-Completeness General Techniques for Combinatorial Approximation Sartaj Sahni epsilon-Approximation Schemes for the Constrained Shortest Path Problem Krishnaiyan "KT" Thulasiraman Constrained Shortest Path Problem: Lagrangian Relaxation-Based Algorithmic Approaches Ying Xiao and Krishnaiyan "KT" Thulasiraman Algorithms for Finding Disjoint Paths with QoS Constraints Alex Sprintson and Ariel Orda Set-Cover Approximation Neal E. Young Approximation Schemes for Fractional Multicommodity Flow Problems George Karakostas Approximation Algorithms for Connectivity Problems Ramakrishna Thurimella Rectilinear Steiner Minimum Trees Tao Huang and Evangeline F.Y. Young Fixed-Parameter Algorithms and Complexity Venkatesh Raman and Saket Saurabh

Abstract: In 1982, Zaslavsky introduced the concept of a proper vertex colouring of a signed graph $G$ as a mapping $\phi\colon V(G)\to \mathbb{Z}$ such that for any two adjacent vertices $u$ and $v$ the colour $\phi(u)$ is different from the colour $\sigma(uv)\phi(v)$, where is $\sigma(uv)$ is the sign of the edge $uv$. The substantial part of Zaslavsky's research concentrated on polynomial invariants related to signed graph colourings rather than on the behaviour of colourings of individual signed graphs. We continue the study of signed graph colourings by proposing the definition of a chromatic number for signed graphs which provides a natural extension of the chromatic number of an unsigned graph. We establish the basic properties of this invariant, provide bounds in terms of the chromatic number of the underlying unsigned graph, investigate the chromatic number of signed planar graphs, and prove an extension of the celebrated Brooks' theorem to signed graphs.

Abstract: We study a general class of problems called $\mathcal{F}$-Deletion problems. In an $\mathcal{F}$-Deletion problem, we are asked whether a subset of at most $k$ vertices can be deleted from a graph $G$ such that the resulting graph does not contain as a minor any graph from the family ${\cal F}$ of forbidden minors. We study the problem parameterized by $k$, using $p$-$\mathcal{F}$-Deletion to refer to the parameterized version of the problem. We obtain a number of algorithmic results on the $p$-$\mathcal{F}$-Deletion problem when $\mathcal{F}$ contains a planar graph. We give a linear vertex kernel on graphs excluding $t$-claw $K_{1,t}$, the star with $t$ leaves, as an induced subgraph, where $t$ is a fixed integer and an approximation algorithm achieving an approximation ratio of $O(\log^{3/2} OPT)$, where $OPT$ is the size of an optimal solution on general undirected graphs. Finally, we obtain polynomial kernels for the case when $\cal F$ only contains graph $\theta_c$ as a minor for a fixed integer $c$...

Abstract: Approximate matchings in fully dynamic graphs have been intensively studied in recent years. Cupta and Peng [FOCS'13] presented a deterministic algorithm for maintaining fully dynamic (1 + e)-approximate maximum cardinality matching (MCM) in general graphs with worst-case update time O([EQUATION]), for any e > 0, where m denotes the current number of edges in the graph. Despite significant research efforts, this [EQUATION] update time barrier remains the state-of-the-art even if amortized time bounds and randomization are allowed or the approximation factor is allowed to increase from 1 + e to 2 -- e, and even in basic graph families such as planar graphs.This paper presents a simple deterministic algorithm whose performance depends on the density of the graph. Specifically, we maintain fully dynamic (1 + e)-approximate MCM with worst-case update time O(α ·e--2) for graphs with arboricity1 bounded by α. The update time bound holds even if the arboricity bound α changes dynamically. Since the arboricity ranges between 1 and [EQUATION], our density-sensitive bound O(α · e--2) naturally generalizes the O([EQUATION] · e--2) bound of Gupta and Peng.For the family of bounded arboricity graphs (which includes forests, planar graphs, and graphs excluding a fixed minor), in the regime e = O(1) our update time reduces to a constant. This should be contrasted with the previous best 2-approximation results for bounded arboricity graphs, which achieve either an O(log n) worst-case bound (Kopelowitz et al., ICALP'14) or an O([EQUATION]) amortized bound (He et al., ISAAC'14), where n stands for the number of vertices in the graph.En route to this result, we provide local algorithms of independent interest for maintaining fully dynamic approximate matching and vertex cover.

Abstract: Fullerene graphs are cubic, 3-connected, planar graphs with exactly 12 pentagonal faces, while all other faces are hexagons. Fullerene graphs are mathematical models of fullerene molecules, i.e., molecules comprised only by carbon atoms different than graphites and diamonds. We give a survey on fullerene graphs from our perspective, which could be also considered as an introduction to this topic. Different types of fullerene graphs are considered, their symmetries, and construction methods. We give an overview of some graph invariants that can possibly correlate with the fullerene molecule stability, such as: the bipartite edge frustration, the independence number, the saturation number, the number of perfect matchings, etc.

Abstract: A known failing of many popular random graph models is that the Aldous-Hoover Theorem guarantees these graphs are dense with probability one; that is, the number of edges grows quadratically with the number of nodes. This behavior is considered unrealistic in observed graphs. We define a notion of edge exchangeability for random graphs in contrast to the established notion of infinite exchangeability for random graphs --- which has traditionally relied on exchangeability of nodes (rather than edges) in a graph. We show that, unlike node exchangeability, edge exchangeability encompasses models that are known to provide a projective sequence of random graphs that circumvent the Aldous-Hoover Theorem and exhibit sparsity, i.e., sub-quadratic growth of the number of edges with the number of nodes. We show how edge-exchangeability of graphs relates naturally to existing notions of exchangeability from clustering (a.k.a. partitions) and other familiar combinatorial structures.

Abstract: We study the topic of “extremal” planar graphs, defining to be the maximum number of edges possible in a planar graph on n vertices that does not contain a given graph H as a subgraph. In particular, we examine the case when H is a small cycle, obtaining for all and for all , and showing that both of these bounds are tight.

Abstract: We give the first polynomial-time approximation schemes (PTASs) for the following problems: (1) uniform facility location in edge-weighted planar graphs, (2) k-median and k-means in edge-weighted planar graphs, (3) k-means in Euclidean space of bounded dimension. Our first and second results extend to minor-closed families of graphs. All our results extend to cost functions that are the pth power of the shortest-path distance. The algorithm is local search where the local neighborhood of a solution S consists of all solutions obtained from S by removing and adding 1/eO(1) centers.

Abstract: In this work we study and expand a model describing the dynamics of a unicellular slime mold, Physarum Polycephalum (PP), which was proposed to simulate the ability of PP to find the shortest path connecting two food sources in a maze. The original model describes the dynamics of the slime mold on a finite dimensional planar graph using a pipe-flow analogy whereby mass transfer occurs because of pressure differences with a conductivity coefficient that varies with the flow intensity. We propose an extension of this model that abandons the graph structure and moves to a continuous domain. Numerical evidence, shows that the model is capable of describing the slime mold dynamics also for large times, accurately reproducing the PP behavior. A notable result related to the original model is that it is equivalent to an optimal transportation problem over the graph as time tends to infinity. In our case, we can only conjecture that our extension presents a time-asymptotic equilibrium. This equilibrium point is precisely the solution of the Monge-Kantorovich (MK) equations at the basis of the PDE formulation of optimal transportation problems. Numerical results obtained with our approach, which combines P1 Finite Elements with forward Euler time stepping, show that the approximate solution converges at large times to an equilibrium configuration that well compares with the numerical solution of the MK-equations.

Abstract: Distributed optimization algorithms are frequently faced with solving sub-problems on disjoint connected parts of a network. Unfortunately, the diameter of these parts can be significantly larger than the diameter of the underlying network, leading to slow running times. Recent work by [Ghaffari and Hauepler; SODA'16] showed that this phenomenon can be seen as the broad underlying reason for the pervasive Omega(√n + D) lower bounds that apply to most optimization problems in the CONGEST model. On the positive side, this work also introduced low-congestion shortcuts as an elegant solution to circumvent this problem in certain topologies of interest. Particularly, they showed that there exist good shortcuts for any planar network and more generally any bounded genus network. This directly leads to fast O(DlogO(1)n) distributed optimization algorithms on such topologies, e.g., for MST and Min-Cut approximation, given that one can efficiently construct these shortcuts in a distributed manner.Unfortunately, the shortcut construction of [Ghaffari and Hauepler; SODA'16] relies heavily on having access to a bounded genus embedding of the network. Computing such an embedding distributedly, however, is a hard problem - even for planar networks. No distributed embedding algorithm for bounded genus graphs is in sight. In this work, we side-step this problem by defining a slightly restricted and more structured form of shortcuts and giving a novel construction algorithm which efficiently finds a shortcut which is, up to a logarithmic factor, as good as the best shortcut that exists for a given network. This new construction algorithm directly leads to an O(D logO(1) n)-round algorithm for solving optimization problems like MST for any topology for which good restricted shortcuts exist - without the need to compute any embedding. This includes the first efficient algorithm for bounded genus graphs.

Abstract: The dynamic shortest paths problem on planar graphs asks us to preprocess a planar graph G such that we may support insertions and deletions of edges in G as well as distance queries between any two nodes u, v subject to the constraint that the graph remains planar at all times. This problem has been extensively studied in both the theory and experimental communities over the past decades. The best known algorithm performs queries and updates in O(n2/3) time, based on ideas of a seminal paper by Fakcharoenphol and Rao [FOCS'01]. A (1+e)-approximation algorithm of Abraham et al. [STOC'12] performs updates and queries in O(√n) time. An algorithm with a more practical O(polylog(n)) runtime would be a major breakthrough. However, such runtimes are only known for a (1+e)-approximation in a model where only restricted weight updates are allowed due to Abraham et al. [SODA'16], or for easier problems like connectivity. In this paper, we follow a recent and very active line of work on showing lower bounds for polynomial time problems based on popular conjectures, obtaining the first such results for natural problems in planar graphs. Such results were previously out of reach due to the highly non-planar nature of known reductions and the impossibility of "planarizing gadgets". We introduce a new framework which is inspired by techniques from the literatures on distance labelling schemes and on parameterized complexity. Using our framework, we show that no algorithm for dynamic shortest paths or maximum weight bipartite matching in planar graphs can support both updates and queries in amortized O(n1/2-e) time, for any e>0, unless the classical all-pairs-shortest-paths problem can be solved in truly subcubic time, which is widely believed to be impossible. We extend these results to obtain strong lower bounds for other related problems as well as for possible trade-offs between query and update time. Interestingly, our lower bounds hold even in very restrictive models where only weight updates are allowed.

Abstract: A graph with n vertices is 1-planar if it can be drawn in the plane such that each edge is crossed at most once, and is optimal if it has the maximum of 4n-8 edges. We show that optimal 1-planar graphs can be recognized in linear time. Our algorithm implements a graph reduction system with two rules, which can be used to reduce every optimal 1-planar graph to an irreducible extended wheel graph. The graph reduction system is non-deterministic, constraint, and non-confluent.

Abstract: We apply the concept of single-valued neutrosophic sets to multigraphs, planar graphs and dual graphs. We introduce the notions of single-valued neutrosophic multigraphs, single-valued neutrosophic planar graphs, and single-valued neutrosophic dual graphs. We illustrate these concepts with examples. We also investigate some of their properties.

Abstract: We exactly settle the complexity of graph realization, graph rigidity, and graph global rigidity as applied to three types of graphs: "globally noncrossing" graphs, which avoid crossings in all of their configurations; matchstick graphs, with unit-length edges and where only noncrossing configurations are considered; and unrestricted graphs (crossings allowed) with unit edge lengths (or in the global rigidity case, edge lengths in {1,2}). We show that all nine of these questions are complete for the class Exists-R, defined by the Existential Theory of the Reals, or its complement Forall-R; in particular, each problem is (co)NP-hard. One of these nine results - that realization of unit-distance graphs is Exists-R-complete - was shown previously by Schaefer (2013), but the other eight are new. We strengthen several prior results. Matchstick graph realization was known to be NP-hard (Eades & Wormald 1990, or Cabello et al. 2007), but its membership in NP remained open; we show it is complete for the (possibly) larger class Exists-R. Global rigidity of graphs with edge lengths in {1,2} was known to be coNP-hard (Saxe 1979); we show it is Forall-R-complete. The majority of the paper is devoted to proving an analog of Kempe's Universality Theorem - informally, "there is a linkage to sign your name" - for globally noncrossing linkages. In particular, we show that any polynomial curve phi(x,y)=0 can be traced by a noncrossing linkage, settling an open problem from 2004. More generally, we show that the nontrivial regions in the plane that may be traced by a noncrossing linkage are precisely the compact semialgebraic regions. Thus, no drawing power is lost by restricting to noncrossing linkages. We prove analogous results for matchstick linkages and unit-distance linkages as well.

Abstract: An $st$-path in a drawing of a graph is self-approaching if during the traversal of the corresponding curve from $s$ to any point $t'$ on the curve the distance to $t'$ is non-increasing. A path has increasing chords if it is self-approaching in both directions. A drawing is self-approaching (increasing-chord) if any pair of vertices is connected by a self-approaching (increasing-chord) path. We study self-approaching and increasing-chord drawings of triangulations and 3-connected planar graphs. We show that in the Euclidean plane, triangulations admit increasing-chord drawings, and for planar 3-trees we can ensure planarity. We prove that strongly monotone (and thus increasing-chord) drawings of trees and binary cactuses require exponential resolution in the worst case, answering an open question by Kindermann et al. (GD 2014). Moreover, we provide a binary cactus that does not admit a self-approaching drawing. Finally, we show that 3-connected planar graphs admit increasing-chord drawings in the hyperbolic plane and characterize the trees that admit such drawings.

Abstract: We present data stream algorithms for estimating the size or weight of the maximum matching in low arboricity graphs. A large body of work has focused on improving the constant approximation factor for general graphs when the data stream algorithm is permitted O(n polylog n) space where n is the number of nodes. This space is necessary if the algorithm must return the matching. Recently, Esfandiari et al. (SODA 2015) showed that it was possible to estimate the maximum cardinality of a matching in a planar graph up to a factor of 24+epsilon using O(epsilon^{-2} n^{2/3} polylog n) space. We first present an algorithm (with a simple analysis) that improves this to a factor 5+epsilon using the same space. We also improve upon the previous results for other graphs with bounded arboricity. We then present a factor 12.5 approximation for matching in planar graphs that can be implemented using O(log n) space in the adjacency list data stream model where the stream is a concatenation of the adjacency lists of the graph. The main idea behind our results is finding "local" fractional matchings, i.e., fractional matchings where the value of any edge e is solely determined by the edges sharing an endpoint with e. Our work also improves upon the results for the dynamic data stream model where the stream consists of a sequence of edges being inserted and deleted from the graph. We also extend our results to weighted graphs, improving over the bounds given by Bury and Schwiegelshohn (ESA 2015), via a reduction to the unweighted problem that increases the approximation by at most a factor of two.

Abstract: We prove a number of double-sided estimates relating discrete counterparts of several classical conformal invariants of a quadrilateral: cross-ratios, extremal lengths and random walk partition functions. The results hold true for any simply connected discrete domain Ω with four marked boundary vertices and are uniform with respect to Ω’s which can be very rough, having many fiords and bottlenecks of various widths. Moreover, due to results from [Boundaries of planar graphs, via circle packings (2013) Preprint], those estimates are fulfilled for domains drawn on any infinite “properly embedded” planar graph Γ⊂C (e.g., any parabolic circle packing) whose vertices have bounded degrees. This allows one to use classical methods of geometric complex analysis for discrete domains “staying on the microscopic level.” Applications include a discrete version of the classical Ahlfors–Beurling–Carleman estimate and some “surgery technique” developed for discrete quadrilaterals.

Abstract: Steinberg conjectured in 1976 that every planar graph with no cycles of length four or five is 3-colorable. We disprove this conjecture.

Abstract: Wegner conjectured in 1977 that the square of every planar graph with maximum degree at most $3$ is $7$-colorable. We prove this conjecture using the discharging method and computational techniques to verify reducible configurations.

Abstract: By now, we have a good understanding of how NP-hard problems become easier on graphs of bounded treewidth and bounded cliquewidth: for various problems, matching upper bounds and conditional lower bounds describe exactly how the running time has to depend on treewidth or cliquewidth. In particular, Fomin et al. (2009, 2010) have shown a significant difference between these two parameters: assuming the Exponential-Time Hypothesis (ETH), the optimal algorithms for problems such as Max Cut and Edge Dominating Set have running time 2O(t)nO(1) when parameterized by treewidth, but nO(t) when parameterized by cliquewidth.In this paper, we show that a similar phenomenon occurs also for counting problems. Specifically, we prove that, assuming the counting version of the Strong Exponential-Time Hypothesis (#SETH), the problem of counting perfect matchings• has no (2 --- e)knO(1) time algorithm for any e > 0 on graphs of treewidth k (but it is known to be solvable in time 2knO(1) if a tree decomposition of width k is given), and• has no O(n(1-e)k) time algorithm for any e > 0 on graphs of cliquewidth k (but it can be solved in time O(nk+1) if a k-expression is given).A celebrated result of Fisher, Kasteleyn, and Temperley from the 1960s shows that counting perfect matchings in planar graphs is polynomial-time solvable. This was later extended by Gallucio and Loebl (1999), Tesler (2000) and Regge and Zechina (2000) who gave 4k · nO(1) time algorithms for graphs of genus k. We show that the dependence on the genus k has to be exponential: assuming #ETH, the counting version of ETH, there is no 2O(k) · nO(1) time algorithm for the problem on graphs of genus k.