Search

This paper is devoted to study the combinatorial properties of Local Minimum Spanning Trees (LMSTs), a geometric structure that is attracting increasing research interest in the wireless sensor networks community. Namely, we study which topologies are allowed for a sensor network that uses, for supporting connectivity, a local minimum spanning tree approach. First, we refine the current definition of LMST realizability, focusing on the role of the power of transmission (i.e., of the radius of the covered area). Second, we show simple planar, connected, and triangle-free graphs with maximum degree 3 that cannot be represented as an LMST. Third, we present several families of graphs that can be represented as LMSTs. Then, we show a relationship between planar graphs and their representability as LMSTs based on homeomorphism. Finally, we show that the general problem of determining whether a graph is LMST representable is NP-hard.

A polynomial-time algorithm for testing if a triconnected directed graph has an upward drkwing is presented. An upward drkwing is a planar drkwing such that all the edges flow in a common direction (e.g., from bottom to top). The problem arises in the fields of automatic graph drkwing and ordered sets, and has been open for several years. The proposed algorithm is based on a new combinatorial characterization that maps the problem into a max-flow problem on a sparse network; the time complexity isO(n+r 2) , wheren is the number of vertices andr is the number of sources and sinks of the directed graph. If the directed graph has an upward drkwing, the algorithm allows us to construct one easily.

In this paper we give a model for dynamic graph algorithms, based on performing queries and updates on an implicit representation of the drawing. We present dynamic algorithms for drawing planar graphs that use a variety of drawing standards (such as polyline, straight-line, orthogonal, grid, upward, and visibility drawings), and address aesthetic criteria that are important for readability, such as the display of planarity, symmetry, and reachability. Also, we provide techniques that are especially tailored for important subclasses of planar graphs such as trees and series-parallel digraphs. Our dynamic drawing algorithms have the important property of performing “smooth updates” of the drawing. Of special geometric interest is the possibility of performing point-location and window queries on the implicit representation of the drawing.

The goal of this paper is to investigate the area requirements for upward grid drawings of binary trees. First, we show that there is a family of binary trees with n vertices that require ω(n log n) area; this bound is tight to within a constant factor, i.e. any binary tree with n vertices can be drawn in O(n log n) area. Then we present an algorithm for constructing an upward drawing of a complete binary tree with n vertices in O(n) area, and, finally, we extend this result to the drawings of Fibonacci trees.

The state of the art in automatic graph drawing is reviewed, with special attention to the readability of information system diagrams. Existing results in the literature are compared, and a comprehensive algorithmic approach to the problem is proposed. The algorithm presented draws graphs on a grid and is suitable for both undirected graphs and mixed graphs that contain as subgraphs hierarchic structures. Several applications of GIOTTO, a graphic tool that embodies the aforementioned facility, are shown. >

In diagrammatic representations of hierarchies the minimization of the number of crossings between edges is a well-known criterion for improving readability. An efficient algorithm for testing if a hierarchy is planar (i.e. if it can be drawn without edge crossings) is proposed. A complete combinatorial characterization of the class of planar hierarchies is also given. >

The incremental planarity testing problem consists of performing the following operations on a planar graph G with n vertices: (1) testing whether a new edge can be added to G so that the resulting graph is itself planar; (2) adding vertices and edges such that planarity is preserved. An efficient technique for incremental planarity testing that uses O(n) space and supports tests and insertion of vertices and edges in O(log n) time is presented. The bounds for queries and vertex insertions are worst case, and the bound for edge insertions is amortized. >

An algorithm for k-line planarity testing of hierarchical graphs is described. If the graph is k-line planar the algorithm produces the corresponding embedding. The algorithm has time complexity O(∣V∣). Furthermore some forbidden patterns are identified, whose existence is necessary and sufficient for a hierarchical graph to be not k-line planar.