34 research outputs found
Patrolling a Street Network is Strongly NP-Complete but in P for Tree Structures
Full text linkWe consider the following problem: Given a finite set of straight line
segments in the plane, determine the positions of a minimal number of points on
the segments, from which guards can see all segments. This problem can be
interpreted as looking for a minimal number of locations of policemen, guards,
cameras or other sensors, that can observe a network of streets, corridors,
tunnels, tubes, etc. We show that the problem is strongly NP-complete even for
a set of segments with a cubic graph structure, but in P for tree structures
Formulas for the number of (n−2)-gaps of binary objects in arbitrary dimension
Get PDFAbstractIn this paper we define the notion of a gap in an arbitrary digital binary object S in a digital space of arbitrary dimension. Then we obtain an explicit formula for the number of gaps in S of maximal dimension, derive combinatorial relations for digital curves, and discuss possible applications to image analysis of digital surfaces (in particular planes) and curves
Formulas for the number of <mml:math altimg="si21.gif" display="inline" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:cals="http://www.elsevier.com/xml/common/cals/dtd"><mml:mrow><mml:mo>(</mml:mo><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:math>-gaps of binary objects in arbitrary dimension
No full textA Class of Fibonacci Matrices, Graphs, and Games
Get PDFIn this paper, we define a class of Fibonacci graphs as graphs whose adjacency matrices are obtained by alternating binary Fibonacci words. We show that Fibonacci graphs are close in size to Turán graphs and that their size-stability tradeoff defined as the product of their size and stability number is very close to the maximum possible over all bipartite graphs. We also consider a combinatorial game based on sequential vertex deletions and show that the Fibonacci graphs are extremal regarding the number of rounds in which the game can terminate
A Class of Fibonacci Matrices, Graphs, and Games
No full textIn this paper, we define a class of Fibonacci graphs as graphs whose adjacency matrices are obtained by alternating binary Fibonacci words. We show that Fibonacci graphs are close in size to Turán graphs and that their size-stability tradeoff defined as the product of their size and stability number is very close to the maximum possible over all bipartite graphs. We also consider a combinatorial game based on sequential vertex deletions and show that the Fibonacci graphs are extremal regarding the number of rounds in which the game can terminate
