Md. Jawaherul Alam

Straight-Line Grid Drawings of 3-Connected 1-Planar Graphs

A graph is 1-planar if it can be drawn in the plane such that each edge is crossed at most once. In general, 1-planar graphs do not admit straight-line drawings. We show that every 3-connected 1-planar graph has a straight-line drawing on an integer grid of quadratic size, with the exception of a single edge on the outer face that has one bend. The drawing can be computed in linear time from any given 1-planar embedding of the graph.

Computing Cartograms with Optimal Complexity

In a rectilinear dual of a planar graph vertices are represented by simple rectilinear polygons, while edges are represented by side-contact between the corresponding polygons. A rectilinear dual is called a cartogram if the area of each region is equal to a pre-specified weight. The complexity of a cartogram is determined by the maximum number of corners (or sides) required for any polygon. In a series of papers the polygonal complexity of such representations for maximal planar graphs has been reduced from the initial 40 to 34, then to 12 and very recently to the currently best known 10. Here we describe a construction with 8-sided polygons, which is optimal in terms of polygonal complexity as 8-sided polygons are sometimes necessary. Specifically, we show how to compute the combinatorial structure and how to refine it into an area-universal rectangular layout in linear time. The exact cartogram can be computed from the area-universal layout with numerical iteration, or can be approximated with a hill-climbing heuristic. We also describe an alternative construction of cartograms for Hamiltonian maximal planar graphs, which allows us to directly compute the cartograms in linear time. Moreover, we prove that even for Hamiltonian graphs 8-sided rectilinear polygons are necessary, by constructing a non-trivial lower bound example. The complexity of the cartograms can be reduced to 6 if the Hamiltonian path has the extra property that it is one-legged, as in outer-planar graphs. Thus, we have optimal representations (in terms of both polygonal complexity and running time) for Hamiltonian maximal planar and maximal outer-planar graphs. Finally we address the problem of constructing small-complexity cartograms for 4-connected graphs (which are Hamiltonian). We first disprove a conjecture, posed by two set of authors, that any 4-connected maximal planar graph has a one-legged Hamiltonian cycle, thereby invalidating an attempt to achieve a polygonal complexity 6 in cartograms for this graph class. We also prove that it is NP-hard to decide whether a given 4-connected plane graph admits a cartogram with respect to a given weight function.

Computing cartograms with optimal complexity

In a rectilinear dual of a planar graph vertices are represented by simple rectilinear polygons, while edges are represented by side-contact between the corresponding polygons. A rectilinear dual is called a cartogram if the area of each region is equal to a pre-specified weight. The complexity of a cartogram is determined by the maximum number of corners (or sides) required for any polygon. In a series of papers the polygonal complexity of such representations for maximal planar graphs has been reduced from the initial 40 to 34, then to 12 and very recently to the currently best known 10. Here we describe a construction with 8-sided polygons, which is optimal in terms of polygonal complexity as 8-sided polygons are sometimes necessary. Specifically, we show how to compute the combinatorial structure and how to refine it into an area-universal rectangular layout in linear time. The exact cartogram can be computed from the area-universal layout with numerical iteration, or can be approximated with a hill-climbing heuristic. We also describe an alternative construction of cartograms for Hamiltonian maximal planar graphs, which allows us to directly compute the cartograms in linear time. Moreover, we prove that even for Hamiltonian graphs 8-sided rectilinear polygons are necessary, by constructing a non-trivial lower bound example. The complexity of the cartograms can be reduced to 6 if the Hamiltonian path has the extra property that it is one-legged, as in outer-planar graphs. Thus, we have optimal representations (in terms of both polygonal complexity and running time) for Hamiltonian maximal planar and maximal outer-planar graphs.

Quantitative Measures for Cartogram Generation Techniques

Cartograms are used to visualize geographically distributed data by scaling the regions of a map (e.g., US states) such that their areas are proportional to some data associated with them (e.g., population). Thus the cartogram computation problem can be considered as a map deformation problem where the input is a planar polygonal map M and an assignment of some positive weight for each region. The goal is to create a deformed map M′, where the area of each region realizes the weight assigned to it (no cartographic error) while the overall map remains readable and recognizable (e.g., the topology, relative positions and shapes of the regions remain as close to those before the deformation as possible). Although several such measures of cartogram quality are well‐known, different cartogram generation methods optimize different features and there is no standard set of quantitative metrics. In this paper we define such a set of seven quantitative measures, designed to evaluate how faithfully a cartogram represents the desired weights and to estimate the readability of the final representation. We then study several cartogram‐generation algorithms and compare them in terms of these quantitative measures.

Threshold-coloring and unit-cube contact representation of planar graphs

Abstract In this paper we study threshold-coloring of graphs, where the vertex colors represented by integers are used to describe any spanning subgraph of the given graph as follows. A pair of vertices with a small difference in their colors implies that the edge between them is present, while a pair of vertices with a big color difference implies that the edge is absent. Not all planar graphs are threshold-colorable, but several subclasses, such as trees, some planar grids, and planar graphs with no short cycles can always be threshold-colored. Using these results we obtain unit-cube contact representation of several subclasses of planar graphs. Variants of the threshold-coloring problem are related to well-known graph coloring and other graph-theoretic problems. Using these relations we show the NP-completeness for two of these variants, and describe a polynomial-time algorithm for another.

Happy edges: Threshold-coloring of regular lattices

We study a graph coloring problem motivated by a fun Sudoku-style puzzle. Given a bipartition of the edges of a graph into near and far sets and an integer threshold t, a threshold-coloring of the graph is an assignment of integers to the vertices so that endpoints of near edges differ by t or less, while endpoints of far edges differ by more than t. We study threshold-coloring of tilings of the plane by regular polygons, known as Archimedean lattices, and their duals, the Laves lattices. We prove that some are threshold-colorable with constant number of colors for any edge labeling, some require an unbounded number of colors for specific labelings, and some are not threshold-colorable.

Threshold-Coloring and Unit-Cube Contact Representation of Graphs

We study threshold coloring of graphs where the vertex colors, represented by integers, describe any spanning subgraph of the given graph as follows. Pairs of vertices with near colors imply the edge between them is present and pairs of vertices with far colors imply the edge is absent. Not all planar graphs are threshold-colorable, but several subclasses, such as trees, some planar grids, and planar graphs with no short cycles can always be threshold-colored. Using these results we obtain unit-cube contact representation of several subclasses of planar graphs. We show the NP-completeness for two variants of the threshold coloring problem and describe a polynomial-time algorithm for another.

Orthogonal layout with optimal face complexity

Abstract We study a problem motivated by rectilinear schematization of geographic maps. Given a biconnected plane graph G and an integer k ≥ 0 , does G have a strict-orthogonal drawing (i.e., an orthogonal drawing without edge bends) with at most k reflex angles per face? For k = 0 , the problem is equivalent to realizing each face as a rectangle. We prove that the strict-orthogonal drawability problem for arbitrary reflex complexity k can be reduced to a graph matching or a network flow problem. Consequently, we obtain an O ˜ ( n 10 / 7 k 1 / 7 ) -time algorithm to decide strict-orthogonal drawability, where O ˜ ( r ) denotes O ( r log c ⁡ r ) , for some constant c. In contrast, if the embedding is not fixed, we prove that it is NP-complete to decide whether a planar graph admits a strict-orthogonal drawing with reflex face complexity 4.

Proportional Contact Representations of Planar Graphs

We study contact representations for planar graphs, with vertices represented by simple polygons and adjacencies represented by point-contacts or side-contacts between the corresponding polygons. Specifically, we consider proportional contact representations, where pre-specified vertex weights must be represented by the areas of the corresponding polygons. Natural optimization goals for such representations include minimizing the complexity of the polygons, and the unused area. We describe algorithms for proportional contact representations with optimal polygonal complexity for general planar graphs and planar 2-segment graphs, which include maximal outer-planar graphs and partial 2-trees. Submitted: November 2011 Reviewed: April 2012 Revised: May 2012 Reviewed: July 2012 Revised: August 2012 Accepted: August 2012 Final: September 2012 Published: September 2012 Article type: Regular paper Communicated by: M. van Kreveld and B. Speckmann Research funded in part by EUROGIGA project GraDR, DFG Fe 340/7-2NSF, NSERC, and NSF grants CCF-0545743 and CCF-1115971. E-mail addresses: mjalam@cs.arizona.edu (Md. Jawaherul Alam) biedl@waterloo.edu (Therese Biedl) felsner@math.tu-berlin.de (Stefan Felsner) mk@informatic.uni-tuebingen.de (Michael Kaufmann) kobourov@cs.arizona.edu (Stephen G. Kobourov) 702 Alam et al. Proportional Contact Representations of Planar Graphs

Contact graphs of circular arcs

We study representations of graphs by contacts of circular arcs, CCA-representations for short, where the vertices are interior-disjoint circular arcs in the plane and each edge is realized by an endpoint of one arc touching the interior of another. A graph is (2, k)-sparse if every s-vertex subgraph has at most \(2s-k\) edges, and (2, k)-tight if in addition it has exactly \(2n-k\) edges, where n is the number of vertices. Every graph with a CCA-representation is planar and (2, 0)-sparse, and it follows from known results that for \(k\ge 3\) every (2, k)-sparse graph has a CCA-representation. Hence the question of CCA-representability is open for (2, k)-sparse graphs with \(0\le k\le 2\). We partially answer this question by computing CCA-representations for several subclasses of planar (2, 0)-sparse graphs. Next, we study CCA-representations in which each arc has an empty convex hull. We show that every plane graph of maximum degree 4 has such a representation, but that finding such a representation for a plane (2, 0)-tight graph with maximum degree 5 is NP-complete. Finally, we describe a simple algorithm for representing plane (2, 0)-sparse graphs with wedges, where each vertex is represented with a sequence of two circular arcs (straight-line segments).