![]() disk contact intersection graphs of parallelograms (squares) unit disk. Note: Many terms used in this article are defined in Glossary of graph theory. A disk graph is the intersection graph of disks in two-dimensional space. Graph coloring is still a very active field of research. It has even reached popularity with the general public in the form of the popular number puzzle Sudoku. Beside the classical types of problems, different limitations can also be set on the graph, or on the way a color is assigned, or even on the color itself. Graph coloring enjoys many practical applications as well as theoretical challenges. The nature of the coloring problem depends on the number of colors but not on what they are. In general, one can use any finite set as the "color set". This improves on 8-approximability as shown by Barrett, Istrate. ![]() In mathematical and computer representations, it is typical to use the first few positive or non-negative integers as the "colors". We show that the strong chromatic index of unit disk graphs is efficiently 6-approximable. By planar duality it became coloring the vertices, and in this form it generalizes to all graphs. This was generalized to coloring the faces of a graph embedded in the plane. The convention of using colors originates from coloring the countries of a map, where each face is literally colored. This is partly pedagogical, and partly because some problems are best studied in their non-vertex form, as in the case of edge coloring. However, non-vertex coloring problems are often stated and studied as-is. For example, an edge coloring of a graph is just a vertex coloring of its line graph, and a face coloring of a plane graph is just a vertex coloring of its dual. Vertex coloring is often used to introduce graph coloring problems, since other coloring problems can be transformed into a vertex coloring instance. We obtain these results using the idea of the logic engine introduced by Bhatt and Cosmadakis in 1987.Methodic assignment of colors to elements of a graph A proper vertex coloring of the Petersen graph with 3 colors, the minimum number possible. Using the reduction we described, we also show that this problem is NP-complete when the given lines are only parallel to the x-axis (and one another). ![]() Disk graphs A (unit) Disk Graph is an intersection graphs of (unit) Disks in the plane. In this paper, we first describe a polynomial-time reduction which shows that deciding whether a graph can be realized as unit disks onto given straight lines is NP-hard, when the given lines are parallel to either the x-axis or y-axis. 1 Chromatic number, independence number and clique number. Based on this scenario, we impose a geometric constraint such that the unit disks must be centered onto given straight lines. We consider the unit disk graph realization problem on a restricted domain, by assuming a scenario where the wireless sensor nodes are deployed on the corridors of a building. It is usually assumed that the nodes have identical sensing ranges, and thus a unit disk graph model is used to model problems concerning wireless sensor networks. One example to such applications is wireless sensor networks, where each disk corresponds to a wireless sensor node, and a pair of intersecting disks corresponds to a pair of sensors being able to communicate with one another. Hence, many researchers attacked this problem by restricting the domain of the disk centers. In some applications, the objects that correspond to unit disks have predefined (geometrical) structures to be placed on. In general, this problem is shown to be $\exists\mathbb\)-complete. ![]() Recognizing unit disk graph is an important geometric problem, and has many application areas. Unit disk graphs are the intersection graphs of unit diameter disks in the Euclidean plane. Our results lead us to propose a conjecture that the class of co-bipartite unit disk graphs is closed under bipartite complementation. In particular, we give structural characterization of those co-bipartite unit disk graphs whose edges between parts form a C4-free bipartite graph, and show that bipartite complements of these graphs are also unit disk graphs. Further we use these results to investigate structure of co-bipartite unit disk graphs. We develop several structural and geometrical tools, and use them to reveal infinitely many new minimal non unit disk graphs. In this paper we study in a systematic way forbidden induced subgraphs for the class of unit disk graphs. We found only finitely many minimal non unit disk graphs in the literature. In spite of quite active study of unit disk graphs very little is known about minimal forbidden induced subgraphs for this class. The class of unit disk graphs is hereditary, and therefore admits a characterization in terms of minimal forbidden induced subgraphs. A unit disk graph is the intersection graph of disks of equal radii in the plane.
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