Md. Saidur Rahman

Rectangular grid drawings of plane graphs

The rectangular grid drawing of a plane graph G is a drawing of G such that each vertex is located on a grid point, each edge is drawn as a horizontal or vertical line segment, and the contour of each face is drawn as a rectangle. In this paper we give a simple linear-time algorithm to find a rectangular grid drawing of G if it exists. We also give an upper bound W + H ≤ n2 on the sum of required width W and height H and a bound W H ≤ n216 on the area of a rectangular grid drawing of G, where n is the number of vertices in G. These bounds are best possible, and hold for any compact rectangular grid drawing.

A linear-time algorithm for four-partitioning four-connected planar graphs

Given a graph G=(V, E), four distinct vertices u1,u2,u3,u4 ∈ V and four natural numbers n1, n2, n3, n4 such that \(\sum\nolimits_{i = 1}^4 {n_i = |V|}\), we wish to find a partition V1, V2, V3, V4 of the vertex set V such that ui ∈ Vi, ¦Vi¦=ni and Vi induces a connected subgraph of G for each i, 1 ≤ i ≤ 4. In this paper we give a simple linear-time algorithm to find such a partition if G is a 4-connected planar graph and u1, u2, u3, u4 are located on the same face of a plane embedding of G. Our algorithm is based on a “4-canonical decomposition” of G, which is a generalization of an st-numbering and a “canonical 4-ordering” known in the area of graph drawings.

Box-Rectangular Drawings of Plane Graphs

In this paper we introduce a new drawing style of a plane graph G called a box-rectangular drawing. It is defined to be a drawing of G on an integer grid such that every vertex is drawn as a rectangle, called a box, each edge is drawn as either a horizontal line segment or a vertical line segment, and the contour of each face is drawn as a rectangle. We establish a necessary and sufficient condition for the existence of a box-rectangular drawing of G. We also give a linear-time algorithm to find a box-rectangular drawing of G if it exists.

Rectangular drawings of planar graphs

A plane graph is a planar graph with a fixed embedding in the plane. In a rectangular drawing of a plane graph, each vertex is drawn as a point, each edge is drawn as a horizontal or vertical line segment, and each face is drawn as a rectangle. A planar graph is said to have a rectangular drawing if at least one of its plane embeddings has a rectangular drawing. In this paper we give a linear-time algorithm to examine whether a planar graph G of maximum degree three has a rectangular drawing or not, and to find a rectangular drawing of G if it exists.

Minimum-Area Drawings of Plane 3-Trees

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 3 − 1⌋ × 2⌈ n 3 ⌉ lower bound for the area of a straight-line grid drawing of a plane 3tree with n ≥ 6 vertices, which improves the previously known lower bound ⌊ 2(n−1) 3 ⌋×⌊ 2(n−1) 3 ⌋ for plane graphs.

On Graphs That Are Not PCGs

Let T be an edge weighted tree and let d min ,d max be two nonnegative real numbers. Then the pairwise compatibility graph (PCG) of T is a graph G such that each vertex of G corresponds to a distinct leaf of T and two vertices are adjacent in G if and only if the weighted distance between their corresponding leaves in T is in the interval [d min ,d max ]. Similarly, a given graph G is a PCG if there exist suitable T,d min ,d max , such that G is a PCG of T. Yanhaona, Bayzid and Rahman proved that there exists a graph with 15 vertices that is not a PCG. On the other hand, Calamoneri, Frascaria and Sinaimeri proved that every graph with at most seven vertices is a PCG. In this paper we construct a graph of eight vertices that is not a PCG, which strengthens the result of Yanhaona, Bayzid and Rahman, and implies optimality of the result of Calamoneri, Frascaria and Sinaimeri. We then construct a planar graph with sixteen vertices that is not a PCG. Finally, we prove a variant of the PCG recognition problem to be NP-complete.

Monotone Grid Drawings of Planar Graphs

A monotone drawing of a planar graph

Point-set embeddings of plane 3-trees

G

No-bend orthogonal drawings of series-parallel graphs

is a planar straight-line drawing of

Bend-Minimum Orthogonal Drawings of Plane 3-Graphs

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