Describes efficient sequential and parallel algorithms for orienting the edges of an undirected planar graph so that each vertex has few outgoing edges. From such an orientation one can test in constant time whether a given edge exists. One consequence is a parallel algorithm to list all subgraphs isomorphic to K3 or K4. More recently this paper has been cited for its applications to scheduling update operations in parallel finite element methods.
The complement of a minimum spanning tree is a maximum spanning tree in the dual graph. By applying this fact we can use a modified form of Sleator and Tarjan's dynamic tree data structure to update the MST in logarithmic time per update.
Characterizes two-terminal series graphs in terms of a tree-like structure in their ear decompositions. Uses this characterization to construct parallel algorithms that recognize these graphs and construct their series-parallel decompositions.
It was known that planar graphs have O(n) subgraphs isomorphic to K3 or K4. That is, K3 and K4 have linear subgraph multiplicity. This paper shows that the graphs with linear subgraph multiplicity in the planar graphs are exactly the 3-connected planar graphs. Also, the graphs with linear subgraph multiplicity in the outerplanar graphs are exactly the 2-connected outerplanar graphs.
More generally, let F be a minor-closed family, and let x be the smallest number such that some complete bipartite graph Kx,y is a forbidden minor for F. Then the x-connected graphs have linear subgraph multiplicity for F, and there exists an (x − 1)-connected graph (namely Kx − 1,x − 1) that does not have linear subgraph multiplicity. When x ≤ 3 or when x = 4 and the minimal forbidden minors for F are triangle-free, then the graphs with linear subgraph multiplicity for F are exactly the x-connected graphs.
Please refer only to the journal version, and not the earlier technical report: the technical report had a bug that was repaired in the journal version.
Replaces portions of a hierarchical separator decomposition with smaller certificates to achieve fast update times for various dynamic planar graph problems. Applications include finding the k best spanning trees of a planar graph.
First half of journal version of Separator based sparsification for dynamic planar graph algorithms.
Second half of journal version of Separator based sparsification for dynamic planar graph algorithms.
Uses an idea of Baker to cover a planar graph with subgraphs of low treewidth. As a consequence, any fixed pattern can be found as a subgraph in linear time; the same methods can be used to solve other planar graph problems including vertex connectivity, diameter, girth, induced subgraph isomorphism, and shortest paths. A companion paper, "Diameter and treewidth in minor-closed graph families", presents a result announced in the conference version of this paper, that exactly characterizes the minor-closed graph families for which the same techniques apply.
Any algorithm that maintains the connected components of a bitmap image must take Omega(log n / log log n) time per change to the image. The problem can be solved in O(log n) time per change using dynamic planar graph techniques. We discuss applications to computer Go and other games.
This paper introduces the diameter-treewidth property (later known as bounded local treewidth): a functional relationship between the diameter of its graph and its treewidth. Previously known results imply that planar graphs have bounded local treewidth; we characterize the other minor-closed families with this property. Specifically, minor-closed family F has bounded local treewidth if and only if there exists an apex graph G that is not in F; an apex graph is a graph that can be made planar by removing a single vertex. The minor-free families that exclude an apex graph (and therefore have bounded local treewidth) include the bounded-genus graphs (for which, as with planar graphs, we show a linear bound for the treewidth as a function of the diameter) and K3,a-free graphs. As a consequence, subgraph isomorphism for subgraphs of bounded size and approximations to several NP-hard optimization problems can be computed efficiently on these graphs, extending previous results for planar graphs.
Some of these results were announced in the conference version of "subgraph isomorphism for planar graphs and related problems" but not included in the journal version. Since its publication, there have been many more works on local treewidth. The class of problems that could be solved quickly on graphs of bounded local treewidth was extended and classified by Frick and Grohe, "Deciding first-order properties of locally tree-decomposable structures", J. ACM 48: 1184–1206, 2001; the proof that bounded local treewidth is equivalent to having an excluded apex minor was simplified, and the dependence of the treewidth on diameter improved, by a subsequent paper of Demaine and Hajiaghayi, "Diameter and treewidth in minor-closed graph families, revisited", Algorithmica 40: 211–215, 2004. The concept of local treewidth is the basis for the theory of bidimensionality, a general framework for fixed-parameter-tractable algorithms and approximation algorithms in minor-closed graph families; for a survey, see Demaine and Hajiaghayi, "The bidimensionality theory and its algorithmic applications", The Computer J. 51: 292–302, 2008.
We consider several variations of the problem of coloring the squares of a quadtree so that no two adjacent squares are colored alike. We give simple linear time algorithms for 3-coloring balanced quadtrees with edge adjacency, 4-coloring unbalanced quadtrees with edge adjacency, and 6-coloring balanced or unbalanced quadtrees with corner adjacency. The number of colors used by the first two algorithms is optimal; for the third algorithm, 5 colors may sometimes be needed.
We give linear-time quasiconvex programming algorithms for finding a Möbius transformation of a set of spheres in a unit ball or on the surface of a unit sphere that maximizes the minimum size of a transformed sphere. We can also use similar methods to maximize the minimum distance among a set of pairs of input points. We apply these results to vertex separation and symmetry display in spherical graph drawing, viewpoint selection in hyperbolic browsing, and element size control in conformal structured mesh generation.
We show that geometric thickness and book thickness are not asymptotically equivalent: for every t, there exists a graph with geometric thickness two and book thickness > t.
We show that thickness and geometric thickness are not asymptotically equivalent: for every t, there exists a graph with thickness three and geometric thickness > t. The proof uses Ramsey-theoretic arguments similar to those in "Separating book thickness from thickness".
We describe a decomposition of graphs embedded on 2-dimensional manifolds into three subgraphs: a spanning tree, a dual spanning tree, and a set of leftover edges with cardinality determined by the genus of the manifold. This tree-cotree decomposition allows us to find efficient data structures for dynamic graphs (allowing updates that change the surface), better constants in bounded-genus graph separators, and efficient algorithms for tree-decomposition of bounded-genus bounded-diameter graphs.
We describe a new method of drawing graphs, based on allowing the edges to be merged together and drawn as "tracks" (similar to train tracks). We present heuristics for finding such drawings based on my previous algorithms for finding maximal bipartite subgraphs, prove that several important families of graphs have confluent drawings, and provide examples of other graphs that can not be drawn in this way.
We show that graphs with maximum degree four have geometric thickness at most two, by partitioning them into degree two subgraphs and applying simultaneous embedding techniques.
We describe a new algorithm, based on graph matching, for subdividing a triangle mesh (without boundary) so that it has a Hamiltonian cycle of triangles, and prove that this subdivision process increases the total number of triangles in the mesh by at most a factor of 3/2. We also prove lower bounds on the increase needed for meshes with and without boundary.
We describe two algorithms for finding planar layouts of partial cubes: one based on finding the minimum-dimension lattice embedding of the graph and then projecting the lattice onto the plane, and the other based on representing the graph as the planar dual to a weak pseudoline arrangement.
Due to editorial mishandling there will be no journal version of this paper: I submitted it to a journal in 2004, the reviews were supposedly sent back to me in 2005, but I didn't receive them and didn't respond to them, leading the editors to assume that I intended to withdraw the submission. Large portions of the paper have since been incorporated into my book Media Theory, making journal publication moot.
We characterize the graphs that can be drawn confluently with a single confluent track that is tree-like except for three-way Delta junctions, as being exactly the distance hereditary graphs. Based on this characterization, we develop efficient algorithms for drawing these graphs.
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 5n/2 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).
We show how to construct a cubic partial cube from any simplicial arrangement of lines or pseudolines in the projective plane. As a consequence, we find nine new infinite families of cubic partial cubes as well as many sporadic examples.
This follows on to our previous paper on using graph matching to cover a triangulated polyhedral model with a single triangle strip by extending the algorithms to models with boundaries. We provide two methods: one is based on using an algorithm for the Chinese Postman problem to find a small set of triangles to split in order to find a perfect matching in the dual mesh, while the other augments the model with virtual triangles to remove the boundaries and merges the strips formed by our previous algorithm on this augmented model. We implement the algorithms and report some preliminary experimental results.
We consider graph drawing algorithms for learning spaces, a type of st-oriented partial cube derived from antimatroids and used to model states of knowledge of students. We show how to draw any st-planar learning space so all internal faces are convex quadrilaterals with the bottom side horizontal and the left side vertical, with one minimal and one maximal vertex. Conversely, every such drawing represents an $st$-planar learning space. We also describe connections between these graphs and arrangements of translates of a quadrant.
We formalize problems of finding large approximately-matching regions of two related but not completely isomorphic quadrilateral meshes, show that these problems are NP-complete, and describe a natural greedy heuristic that is guaranteed to find good matches when the mismatching parts of the meshes are small.
A partition of a rectangle into smaller rectangles is "area-universal" if any vector of areas for the smaller rectangles can be realized by a combinatorially equivalent partition. These partitions may be applied, for instance, to cartograms, stylized maps in which the shapes of countries have been distorted so that their areas represent numeric data about the countries. We characterize area-universal layouts, describe algorithms for finding them, and discuss related problems. The algorithms for constructing area-universal layouts are based on the distributive lattice structure of the set of all layouts of a given dual graph.
Merged with "Orientation-constrained rectangular layouts" to form the journal version, "Area-universal and constrained rectangular layouts".
We show how to find a stylized map in which regions have been replaced by rectangles, preserving adjacencies between regions, with constraints on the orientations of adjacencies between regions. For an arbitrary dual graph representing a set of adjacencies, and an arbitrary set of orientation constraints, we can determine whether there exists a rectangular map satisfying those constraints in polynomial time. The algorithm is based on a representation of the set of all layouts for a given dual graph as a distributive lattice, and on Birkhoff's representation theorem for distributive lattices.
Merged with "Area-universal rectangular layouts" to form the journal version, "Area-universal and constrained rectangular layouts".
Characterizes squaregraphs as duals of triangle-free hyperbolic line arrangements, provides a forbidden subgraph characterization of them, describes an algorithm for finding minimum subsets of vertices that generate the whole graph by medians, and shows that they may be isometrically embedded into Cartesian products of five (but not, in general, fewer than five) trees.
We consider drawings of planar partial cubes in which all interior faces are centrally symmetric convex polygons, as in my previous paper Algorithms for Drawing Media. Among all drawings of this type, we show how to find the one with optimal angular resolution. The solution involves a transformation from the problem into the parametric negative cycle detection problem: given a graph in which the edge weights are linear functions of a parameter λ, find the minimum value of λ for which the graph contains no negative cycles.
We provide a graph-theoretic characterization of three classes of nonconvex polyhedra with axis-parallel sides, analogous to Steinitz's theorem characterizing the graphs of convex polyhedra.
The journal version has the slightly different title "Steinitz theorems for simple orthogonal polyhedra".
We survey regular labelings for straight-line embedding of planar graphs on grids, rectangular partitions, and orthogonal polyhedra, and the many similarities between these different types of labeling.
Tutte's method of spring embeddings allows any triangulated planar graph to be drawn so that the outer face has any pre-specified convex shape, but it may place vertices exponentially close to each other. Alternative graph drawing methods provide polynomial-area straight line drawings but do not allow the outer face shape to be specified. We describe a drawing method that combines both properties: it has polynomial area, and can match any pre-specified shape of the outer face, even a shape in which some of the vertices have 180 degree angles. We apply our results to drawing polygonal schemas for graphs embedded on surfaces of positive genus.
We show that a partial order has a non-crossing upward planar drawing if and only if it has order dimension two, and we use the Dedekind-MacNeille completion to find a drawing with the minimum possible number of confluent junctions.
We extend Lombardi drawing (in which each edge is a circular arc and the edges incident to a vertex must be equally spaced around it) to drawings in which edges are composed of multiple arcs, and we investigate the graphs that can be drawn in this more relaxed framework.
We show that every planar graph of maximum degree three has a planar Lombardi drawing and that some but not all 4-regular planar graphs have planar Lombardi drawings. The proof idea combines circle packings with a form of Möbius-invariant power diagram for circles, defined using three-dimensional hyperbolic geometry.
For the journal version, see "A Möbius-invariant power diagram and its applications to soap bubbles and planar lombardi drawing.".
A graph is 1-planar if it can be drawn in the plane with at most one crossing per edge, and maximal 1-planar if it is 1-planar but adding any edge would force more than one crossing on some edge or edges. Although maximal 1-planar graphs on n vertices may have as many as 4n − 8 edges, we show that there exist maximal 1-planar graphs with as few as 45n/17 + O(1) edges.
We characterize the graphs of two-dimensional soap bubble clusters as being exactly the bridgeless 3-regular planar graphs. The proof uses the Möbius invariance of the properties characterizing these clusters together with our previous circle packing method for constructing Lombardi drawings of graphs.
For the journal version, see "A Möbius-invariant power diagram and its applications to soap bubbles and planar lombardi drawing.".
We show that testing whether a graph is 1-planar (drawable with at most one crossing per edge) may be performed in polynomial and fixed-parameter tractable time for graphs of bounded circuit rank, vertex cover number, or tree-depth. However, it is NP-complete for graphs of bounded treewidth, pathwidth, or bandwidth.
For every positive integer n, there exists a set of n points on a parabola, with the property that every n-vertex planar graph can be drawn without crossings with its vertices at these points and with its edges drawn as circular arcs.
A superpattern of a set of permutations is a permutation that contains as a pattern every permutation in the set. Previously superpatterns had been considered for all permutations of a given length; we generalize this to sets of permutations defined by forbidden patterns; we show that the 213-avoiding permutations have superpatterns half the length of the known bound for all permutations, and that any proper permutation subclass of the 213-avoiding permutations has near-linear superpatterns. We apply these results to the construction of universal point sets, sets of points that can be used as the vertices of drawings of all n-vertex planar graphs. We use our 213-avoiding superpatterns to construct universal sets of size approximately n2/4, and we also construct near-linear universal sets for graphs of bounded pathwidth.
The planarity game involves rearranging a scrambled line arrangement graph to make it planar. We show that the resulting graphs have drawings in grids of area n7/6, much smaller than the quadratic size bound for grid drawings of planar graphs, and we provide a strategy for planarizing these graphs that is simple enough for human puzzle solving.
A confluent drawing of a graph is a set of points and curves in the plane with the property that two vertices are adjacent in the graph if and only if the corresponding points can be connected to each other by smooth paths in the drawing. We define a variant of confluent drawing, strict confluent drawing, in which a smooth path between two vertices (if it exists) must be unique. We show that it is NP-complete to test whether such drawings exist, in contrast to unrestricted confluence for which the complexity remains open. Additionally, we show that finding outerplanar drawings (in which the points are on the boundary of a disk and the curves are interior to it) with a fixed cyclic vertex ordering can be performed in polynomial time. We use circle packings to find nice versions of these drawings in which all tracks are represented by piecewise-circular curves.
We investigate the number of vertices that must be deleted from an arbitrary graph, in the worst case as a function of the number of edges, in order to planarize the remaining graph. We show that m/5.22 vertices are sufficient and m/(6 − o(1)) are necessary, and we also give bounds for the number of deletions needed to achieve certain subclasses of planar graphs.
The balanced circle packings of the title are systems of interior-disjoint circles, whose tangencies represent the given graph, and whose radii are all within a polynomial factor of each other. We show that these packings always exist for trees, cactus graphs, outerpaths, k-outerplanar graphs of bounded degree when k is at most logarithmic, and planar graphs of bounded treedepth. The treedepth result uses a new construction of inversive-distance circle packings.
Given a plane graph with fixed edge lengths, and an assignment of the angles 0, 180, and 360 to the angles between adjacent edges, we show how to test whether the angle assignment can be realized by an embedding of the graph as a flat folding on a line. As a consequence, we can determine whether two-dimensional cell complexes with one vertex can be flattened. The main idea behind the result is to show that each face of the graph can be folded independently of the other faces.
We give the first known near-linear algorithms for constructing Gomory–Hu trees of bounded-genus graphs, and we shave a log off the time for the same problem on planar graphs.
We study the graphs formed by non-crossing circular arcs in the plane, having a vertex for each arc and an edge for each point where an arc endpoint touches the interior of another arc.
We describe and implement a simple linear time algorithm for recognizing Halin graphs based on two simplifications of triples of degree-three vertices in such graphs. Removing some auxiliary data from the algorithm causes it to recognize a broader class of graphs that we call the D3-reducible graphs. We study the properties of these graphs, showing that they share many properties with the Halin graphs.
We describe a class of polytopes of varying dimensions, whose restriction to three dimensions is the class of roofless polyhedra (Halin graphs). We call these polytopes treetopes. We show that the four-dimensional treetopes are closely related to clustered planar graph drawings, and we use this connection to recognize the graphs of four-dimensional treetopes in polynomial time.
We study the problem of splitting the vertices of a given graph into a bounded number of sub-vertices (with each edge attaching to one of the sub-vertices) in order to make the resulting graph planar. It is NP-complete, but can be approximated to within a constant factor, and is fixed-parameter tractable in the treewidth.
We cluster road networks (modeled as planar graphs, or more generally as graphs obeying a separator theorem) with a given set of cluster centers, by matching graph vertices to centers stably according to distance: no unmatched vertex and center should have smaller distances than the matched pairs for the same points. We provide a separator-based data structure for dynamic nearest neighbor queries in planar or separated graphs, which allows the optimal stable clustering to be constructed in time O(n3/2log n). We also experiment with heuristics for fast practical construction of this clustering.
A penny graph is the contact graph of unit disks: each disk represents a vertex of the graph, no two disks can overlap, and each tangency between two disks represents an edge in the graph. We prove that, when this graph is triangle free, its degeneracy is at most two. As a consequence, triangle-free penny graphs have list chromatic number at most three. We also show that the number of edges in any such graph is at most 2n − Ω(√n).
We study what happens to nonplanar graphs of low width (for various width measures) when they are made planar by replacing crossings by vertices. For treewidth, pathwidth, branchwidth, clique-width, and tree-depth, this replacement can blow up the width from constant to linear. However, for bandwidth, cutwidth, and carving width, graphs of bounded width stay bounded when we planarize them.
We show that the K1,1,3-free partial 2-trees and the Halin graphs other than K4 can all be represented as proper contact graphs of squares in the plane. Among partial 2-trees and Halin graphs, these are exactly the ones that can be embedded without nonempty triangles, which form an obstacle to the existence of square contact representations. However the graph of a square antiprism has no such representation despite being embeddable without any nonempty triangles.
We extend Anari and Vazirani's parallel algorithm for perfect matching in planar graphs to the graph families with a forbidden minor with crossing number one, by developing a concept of mimicking networks for perfect matching.
Clustered planarity is the problem of simultaneously drawing a planar graph and a clustering of its vertices (as Jordan curves surrounding the cluster) with no unnecessary crossings between edges or cluster boundaries; it remains unknown whether it can be solved in polynomial time. We provide parameterized and subexponential exact algorithms for the case where the planar embedding is fixed and the clusters form a partition of the vertices.
Graph Theory – Publications – David Eppstein – Theory Group – Inf. & Comp. Sci. – UC Irvine
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