Research Interests:
A point-set embedding of a planar graph G with n vertices on a set P of n points in R, d ≥ 1, is a straight-line drawing of G, where the vertices of G are mapped to distinct points of P . The problem of computing a point-set embedding of... more
A point-set embedding of a planar graph G with n vertices on a set P of n points in R, d ≥ 1, is a straight-line drawing of G, where the vertices of G are mapped to distinct points of P . The problem of computing a point-set embedding of G on P is NP-complete in R, even when G is 2-outerplanar and the points are in general position. On the other hand, if the points of P are in general position in R, then any bijective mapping of the vertices of G to the points of P determines a point-set embedding of G on P . In this paper, we give an O(n)expected time algorithm to decide whether a plane 3-tree with n vertices admits a point-set embedding on a given set of n points in general position in R and compute such an embedding if it exists, for any fixed ǫ>0. We extend our algorithm to embed a subclass of 4-trees on a point set in R in the form of nested tetrahedra. We also prove that given a plane 3-tree G with n vertices, a set P of n points in R that are not necessarily in general pos...
A crease pattern is an embedded planar graph on a piece of paper. An m×n map is a rectangular piece of paper with a crease pattern that partitions the paper into an m×n regular grid of unit squares. If a map has a configuration such that... more
A crease pattern is an embedded planar graph on a piece of paper. An m×n map is a rectangular piece of paper with a crease pattern that partitions the paper into an m×n regular grid of unit squares. If a map has a configuration such that all the faces of the map are stacked on a unit square and the paper does not self-intersect, then it is flat foldable, and the linear ordering of the faces is called a valid linear ordering. Otherwise, the map is unfoldable. In this paper, we show that given a linear ordering of the faces of an m×n map, we can decide in linear time whether it is a valid linear ordering, which improves the quadratic time algorithm of Morgan. We also define a class of unfoldable 2 × n mountain-valley patterns for every n ≥ 5.
Research Interests:
A grid graph is a finite embedded subgraph of the infinite integer grid. A solid grid graph is a grid graph without holes, i.e., each bounded face of the graph is a unit square. The reconfiguration problem for Hamiltonian cycle or path in... more
A grid graph is a finite embedded subgraph of the infinite integer grid. A solid grid graph is a grid graph without holes, i.e., each bounded face of the graph is a unit square. The reconfiguration problem for Hamiltonian cycle or path in a sold grid graph G asks the following question: given two Hamiltonian cycles (or paths) of G, can we transform one cycle (or path) to the other using some “operation” such that we get a Hamiltonian cycle (or path) of G in the intermediate steps (i.e., after each application of the operation)? In this thesis, we investigate reconfiguration problems for Hamiltonian cycles and paths in the context of two types of solid graphs: rectangular grid graphs, which have a rectangular outer boundary, and L-shaped grid graphs, which have a single reflex corner on the outer boundary, under three operations we define, flip, transpose and switch, that are local in the grid. Reconfiguration of Hamiltonian cycles and paths in embedded grid graphs has potential appl...
Research Interests:
Research Interests: Computer Science and Grid
Research Interests: Computer Science and IEEE
Research Interests:
Abstract. We examine the problem of counting the number of Hamiltonian paths and Hamiltonian cycles in outerplanar graphs and planar graphs, respectively. We give an O (nαn) upper bound and an Ω (αn) lower bound on the maximum number of... more
Abstract. We examine the problem of counting the number of Hamiltonian paths and Hamiltonian cycles in outerplanar graphs and planar graphs, respectively. We give an O (nαn) upper bound and an Ω (αn) lower bound on the maximum number of Hamiltonian ...
Research Interests:
Research Interests:
ABSTRACT In this paper we study bichromatic point-set embeddings of 2-colored trees on 2-colored point sets, i.e., point-set embeddings of trees (whose vertices are colored red and blue) on point sets (whose points are colored red and... more
ABSTRACT In this paper we study bichromatic point-set embeddings of 2-colored trees on 2-colored point sets, i.e., point-set embeddings of trees (whose vertices are colored red and blue) on point sets (whose points are colored red and blue) such that each red (blue) vertex is mapped to a red (resp. blue) point. We prove that deciding whether a given 2-colored tree admits a bichromatic point-set embedding on a given convex point set is an $\cal NP$-complete problem; we also show that the same problem is linear-time solvable if the convex point set does not contain two consecutive points with the same color. Furthermore, we prove a 3n/2−O(1) lower bound and a 2n upper bound (a 7n/6−O(logn) lower bound and a 4n/3 upper bound) on the minimum size of a universal point set for straight-line bichromatic embeddings of 2-colored trees (resp. 2-colored binary trees). Finally, we show that universal convex point sets with n points exist for 1-bend bichromatic point-set embeddings of 2-colored trees.
ABSTRACT A touching triangle graph (TTG) representation of a planar graph is a planar drawing Γ of the graph, where each vertex is represented as a triangle and each edge e is represented as a side contact of the triangles that correspond... more
ABSTRACT A touching triangle graph (TTG) representation of a planar graph is a planar drawing Γ of the graph, where each vertex is represented as a triangle and each edge e is represented as a side contact of the triangles that correspond to the end vertices of e. We call Γ a proper TTG representation if Γ determines a tiling of a triangle, where each tile corresponds to a distinct vertex of the input graph. In this paper we prove that every 3-connected cubic planar graph admits a proper TTG representation. We also construct proper TTG representations for parabolic grid graphs and the graphs determined by rectangular grid drawings (e.g., square grid graphs). Finally, we describe a fixed-parameter tractable decision algorithm for testing whether a 3-connected planar graph admits a proper TTG representation.
Research Interests:
Research Interests:
Research Interests:
ABSTRACT A straight-line drawing of a planar graph G is a planar drawing of G such that each vertex is mapped to a point on the Euclidean plane, each edge is drawn as a straight line segment, and no two edges intersect except possibly at... more
ABSTRACT A straight-line drawing of a planar graph G is a planar drawing of G such that each vertex is mapped to a point on the Euclidean plane, each edge is drawn as a straight line segment, and no two edges intersect except possibly at a common endpoint. A segment in a straight-line drawing is a maximal set of edges that form a straight line segment. A k-segment drawing of G is a straight-line drawing of G such that the number of segments is at most k. A plane graph is a fixed planar embedding of a planar graph. In this paper we prove that it is NP-hard to determine whether a plane graph G with maximum degree four has a k-segment drawing, where ≥3. The problem remains NP-hard when the drawing is constrained to be convex. We also prove that given a partial drawing Γ of a plane graph G, it is NP-hard to determine whether there exists a k-segment drawing of G that contains all the segments specified in Γ, even when G is outerplanar. The problem remains NP-hard for planar graphs with maximum degree three in ℝ 3 when given subsets of the vertices are restricted to be coplanar. Finally, we investigate a worst-case lower bound on the number of segments required by straight-line drawings of arbitrary spanning trees of a given planar graph.
Research Interests:
Research Interests:
Abstract. An acyclic k-coloring of a graph G is a mapping φ from the set of vertices of G to a set of k distinct colors such that no two adjacent vertices receive the same color and φ does not contain any bichromatic cycle. In this paper... more
Abstract. An acyclic k-coloring of a graph G is a mapping φ from the set of vertices of G to a set of k distinct colors such that no two adjacent vertices receive the same color and φ does not contain any bichromatic cycle. In this paper we prove that every triangulated plane ...
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
A straight-line drawing of a planar graph G is a planar drawing of G such that each vertex is mapped to a point on the Euclidean plane, each edge is drawn as a straight line segment, and no two edges intersect except possibly at a common... more
A straight-line drawing of a planar graph G is a planar drawing of G such that each vertex is mapped to a point on the Euclidean plane, each edge is drawn as a straight line segment, and no two edges intersect except possibly at a common endpoint. A segment in a straight-line drawing is a maximal set of edges that form a straight line segment. A k-segment drawing of G is a straight-line drawing of G such that the number of segments is at most k. A plane graph is a fixed planar embedding of a planar graph. In this paper we prove that it is NP-hard to determine whether a plane graph G with maximum degree four has a k-segment drawing, where k � 3. The problem remains NP-hard when the drawing is constrained to be convex. We also prove that given a partial drawing of a plane graph G, it is NP-hard to determine whether there exists a k-segment drawing of G that contains all the segments specified in , even when G is outerplanar. The problem remains NP-hard for planar graphs with maximum d...
Research Interests:
A straight-line grid drawing of a plane graph G is a planar drawing of G, where each vertex is drawn at a grid point of an integer grid and each edge is drawn as a straight-line segment. The height, width and area of such a drawing are... more
A straight-line grid drawing of a plane graph G is a planar drawing of G, where each vertex is drawn at a grid point of an integer grid and each edge is drawn as a straight-line segment. The height, width and area of such a drawing are respectively the height, width and area of the smallest axis-aligned rectangle on the grid which encloses the drawing. A minimum-area drawing of a plane graph G is a straight-line grid drawing of G where the area is the minimum. It is NP-complete to determine whether a plane graph G has a straight-line grid drawing with a given area or not. In this paper we give a polynomial-time algorithm for finding a minimum-area drawing of a plane 3-tree. Furthermore, we show a ⌊ 2n 3 −1⌋×2⌈n ⌉ lower bound for the area of a straight-line grid drawing of 3 a plane 3-tree with n ≥ 6 vertices, which improves the previously known lower bound ⌊ 2(n−1) 3
Abstract. A table cartogram of a two dimensional m n table A of non-negative weights in a rectangle R, whose area equals the sum of the weights, is a partition of R into convex quadrilateral faces corresponding to the cells of A such that... more
Abstract. A table cartogram of a two dimensional m n table A of non-negative weights in a rectangle R, whose area equals the sum of the weights, is a partition of R into convex quadrilateral faces corresponding to the cells of A such that each face has the same adjacency as its cor-responding cell and has area equal to the cell’s weight. Such a partition acts as a natural way to visualize table data arising in various fields of research. In this paper, we give a O(mn)-time algorithm to find a table cartogram in a rectangle. We then generalize our algorithm to obtain ta-ble cartograms inside arbitrary convex quadrangles, circles, and finally, on the surface of cylinders and spheres. 1
A straight-line grid drawing of a plane graph G is a planar drawing of G, where each vertex is drawn at a grid point of an integer grid and each edge is drawn as a straight-line segment. The area of such a drawing is the area of the... more
A straight-line grid drawing of a plane graph G is a planar drawing of G, where each vertex is drawn at a grid point of an integer grid and each edge is drawn as a straight-line segment. The area of such a drawing is the area of the smallest axis-aligned rectangle on the grid which encloses the drawing. A minimum-area drawing of a plane graph G is a straight-line grid drawing of G where the area of the drawing is the minimum. Although it is NP-hard to find minimum-area drawings for general plane graphs, in this paper we obtain minimumarea drawings for plane 3-trees in polynomial time. Furthermore, we show a ⌊2n n 3 − 1 ⌋ × 2⌈ 3 ⌉ lower bound for the area of a straight-line grid drawing of a plane 3-tree with n ≥ 6 vertices, which improves the previously known lower bound ⌊ 2(n−1)
Given a pair of 1-complex Hamiltonian cycles C and \(C'\) in an L-shaped grid graph G, we show that one is reachable from the other under two operations, flip and transpose, while remaining in the family of 1-complex Hamiltonian... more
Given a pair of 1-complex Hamiltonian cycles C and \(C'\) in an L-shaped grid graph G, we show that one is reachable from the other under two operations, flip and transpose, while remaining in the family of 1-complex Hamiltonian cycles throughout the reconfiguration. Operations flip and transpose are local in G. We give a reconfiguration algorithm that uses O(|G|) operations.
Research Interests:
Research Interests:
An acyclic coloring of a graph G is an assignment of colors to the vertices of G such that no two adjacent vertices receive the same color and every cycle in G has vertices of at least three different colors. An acyclic k-coloring of G is... more
An acyclic coloring of a graph G is an assignment of colors to the vertices of G such that no two adjacent vertices receive the same color and every cycle in G has vertices of at least three different colors. An acyclic k-coloring of G is an acyclic coloring of G with at most k colors. It is NP-complete to decide whether a planar graph G with maximum degree four admits an acyclic 3-coloring [1]. Let (u, v) be an edge of G. Subdividing edge (u, v) is the operation of replacing the edge with a path u,w1, . . . , wx, v, where each wi, 1 ≤ i ≤ x, is a vertex of degree two. We call each vertex wi, 1 ≤ i ≤ x, a division vertex. A graph G′ is a subdivision of another graph G, if G′ is obtained by subdividing some edges of G. Wood proved that every graph has a subdivision with two division vertices per edge that is acyclically 3-colorable [5]. Angelini and Frati showed that every triangulated planar graph with n vertices has a subdivision with one division vertex per edge that is acyclicall...
In this paper we study bichromatic point-set embeddings of 2-colored trees on 2-colored point sets, i.e., point-set embeddings of trees (whose vertices are colored red and blue) on point sets (whose points are colored red and blue) such... more
In this paper we study bichromatic point-set embeddings of 2-colored trees on 2-colored point sets, i.e., point-set embeddings of trees (whose vertices are colored red and blue) on point sets (whose points are colored red and blue) such that each red (blue) vertex is mapped to a red (resp. blue) point. We prove that deciding whether a given 2-colored tree admits a bichromatic point-set em- bedding on a given convex point set is an NP-complete problem; we also show that the same problem is linear-time solvable if the convex point set does not contain two consecutive points with the same color. Furthermore, we prove a 3n/2 � O(1) lower bound and a 2n upper bound (a 7n/6 � O(logn) lower bound and a 4n/3 upper bound) on the minimum size of a universal point set for straight-line bichromatic embeddings of 2-colored trees (resp. 2-colored binary trees). Finally, we show that universal convex point sets with n points exist for 1-bend bichromatic point-set embeddings of 2-colored trees.