In 2007, Arkin et al. [3] initiated a systematic study of the complexity of the Hamiltonian cycle problem on square, triangular, or hexagonal grid graphs, restricted to polygonal, thin, superthin, degree-bounded, or solid grid graphs. They solved many combinations of these problems, proving them either polynomially solvable or NP-complete, but left three combinations open. In this paper, we pro...

In the cops and robber game, two players play alternately by moving their tokens along the edges of a graph. The first one plays with the cops and the second one with one robber. The cops aim at capturing the robber, while the robber tries to infinitely evade the cops. The main problem consists in minimizing the number of cops used to capture the robber in a graph. This minimum number is called...

We prove the following graph coloring result Let G be a connected bipartite planar graph Then one can triangulate G in such a way that the resulting graph is colorable This result implies several new upper bounds for guarding problems including the rst non trivial upper bound for the rectilinear Prison Yard Problem n vertex guards are su cient to watch the interior of a rectilinear polygon with...

We connect two aspects of graph drawing, namely angular resolution, and the possibility to draw with all angles an integer multiple of 2π/d. A planar graph with angular resolution at least π/2 can be drawn with all angles an integer multiple of π/2 (rectilinear). For d = 4, d > 2, an angular resolution of 2π/d does not imply that the graph can be drawn with all angles an integer multiple of 2π/...

In geographic information systems and in the production of digital maps for small devices with restricted computational resources one often wants to round coordinates to a rougher grid. This removes unnecessary detail and reduces space consumption as well as computation time. This process is called snapping to the grid and has been investigated thoroughly from a computational-geometry perspecti...

We will prove a d-dimensional version of the Geelen and Simpson theorem. c © 2000 Elsevier Science B.V. All rights reserved.

This paper proposes an organized generalization of Newman and Girvan’s modularity measure for graph clustering. Optimized via a deterministic annealing scheme, this measure produces topologically ordered graph partitions that lead to faithful and readable graph representations on a 2 dimensional SOM like planar grid.

The aim of this work is to investigate the F-index and F-coindex of the line graphs of the cycle graphs, star graphs, tadpole graphs, wheel graphs and ladder graphs using the subdivision concepts. F-index of the line graph of subdivision graph of square grid graph, 2D-lattice, nanotube and nanotorus of TUC4C8[p, q] are also investigated here.

A three-dimensional straight-line grid drawing of a graph, henceforth called a 3D drawing, represents the vertices by distinct grid-points in Z and represents each edge by the line segment between its end vertices, such that no two edges cross. In contrast to the case in the plane, it is folklore that every graph has a 3D drawing. For example, the “moment curve” algorithm places the i th vertex...

A track layout of a graph consists of a vertex colouring, an edge colouring, and a total order of each vertex colour class such that between each pair of vertex colour classes, there is no monochromatic pair of crossing edges. This paper studies track layouts and their applications to other models of graph layout. In particular, we improve on the results of Enomoto and Miyauchi [SIAM J. Discret...