A Fary grid drawing of a graph is a drawing on a three-dimensional grid such that vertices are placed at integer coordinates and edges are straight-lines such that no edge crossings are allowed. In this paper it is proved that each k-colorable graph (k 2) needs at least (n 3=2) volume to be drawn. Furthermore, it is shown how to draw 2-, 3-and 4-colorable graphs in a Fary grid fashion in O(n 2)...

Let G = (V;E) be an arbitrary undirected source graph to be embedded in a target graph EM , the extended grid with vertices on integer grid points and edges to nearest and next-nearest neighbours. We present an algorithm showing how to embed G into EM in both time and space O(jV j) using the new notions of islands and bridges. An island is a connected subgraph in the target graph which is mappe...

A three-dimensional grid drawing of a graph is a placement of the vertices at distinct points with integer coordinates, such that the straight line-segments representing the edges are pairwise non-crossing. A O(n3/2) volume bound is proved for three-dimensional grid drawings of graphs with bounded degree, graphs with bounded genus, and graphs with no bounded complete graph as a minor. The previ...

We consider the minimum feedback vertex set problem for some bipartite graphs and degree-constrained graphs. We show that the problem is linear time solvable for bipartite permutation graphs and NP-hard for grid intersection graphs. We also show that the problem is solvable in O(n2 log6 n) time for n-vertex graphs with maximum degree at most three. key words: 3-regular graph, bipartite permutat...

In this paper, we deal with two-dimensional rectangular arrays for representing row and column structures such as symbol tables, spreadsheets, etc. Firstly we formalize the two-dimensional regular languages by pairs of regular expressions for rows and columns. Next we also introduce context-sensitive graph grammars called labeled grid graph grammars, which characterize the two-dimensional regul...

We study bipartite geometric intersection graphs from the perspective of order dimension. We show that partial orders of height two whose comparability graph is a grid intersection graph have order dimension at most four. Starting from this observation we look at various classes of graphs between grid intersection graph and bipartite permutations graphs and the containment relation on these cla...

Hartman I.B.-A., I. Newman and R. Ziv, On grid intersection graphs, Discrete Mathematics 87 (1991) 41-52. A bipartite graph G = (X, Y; E) has a grid representation if X and Y correspond to sets of horizontal and vertical segments in the plane, respectively, such that (xi, y,) E E if and only if segments x, and y, intersect. We prove that all planar bipartite graphs have a grid representation, a...

Robust control and maintenance of the grid relies on accurate data. Both PMUs and state estimators are prone to false data injection attacks. Thus, it is crucial to have a mechanism for fast and accurate detection of an agent maliciously tampering with the data—for both preventing attacks that may lead to blackouts, and for routine monitoring and control tasks of current and future grids. We pr...

A global multi-grid/GMRES solution methodology on distributed memory machines is successfully developed in this study. To preserve the effectiveness of the multigrid scheme, the grid partitioning is based on the communication graph of the coarsest grid, so that all levels of the multi-grids are located in the same zone (processor). Each node of the graph is weighted with the total number of the...

Text coherence evaluation becomes a vital and lovely task in Natural Language Processing subfields, such as text summarization, question answering, text generation and machine translation. Existing methods like entity-based and graph-based models are engaging with nouns and noun phrases change role in sequential sentences within short part of a text. They even have limitations in global coheren...