scholarly journals An Embedding Algorithm for a Specialcase of Extended Grids

2013 ◽  
Vol 5 ◽  
pp. 40-45
Author(s):  
Mahavir Banukumar
Keyword(s):  
Half Plane ◽  
Page Number ◽  
Linear Time ◽  
Dimensional Space ◽  
Time Algorithm ◽  
Linear Ordering ◽  
Linear Time Algorithm ◽  
3 Dimensional ◽  
Book Embedding ◽  
Minimum Number

A book consists of a line in the 3-dimensional space, called the spine, and a number of pages, each a half-plane with the spine as boundary. A book embedding (π,ρ) of a graph consists of a linear ordering of π, of vertices, called the spine ordering, along the spine of a book and an assignment ρ, of edges to pages so that edges assigned to the same page can be drawn on that page without crossing. That is, we cannot find vertices u, v, x, y with π(u) < π(x) < π(v) < π(y), yet the edges uv and xy are assigned to the same page, that is ρ(uv) = ρ(xy). The book thickness or page number of a graph G is the minimum number of pages in required to embed G in a book. In this paper we consider the extended grid and prove that the 1xn extended grid can be embedded in two pages. We also give a linear time algorithm to embed the 1xn extended grid in two pages.

scholarly journals Embedding Sun Graphs in a Single Page

2013 ◽  
Vol 5 ◽  
pp. 31-36
Author(s):  
Mahavir Banukumar
Keyword(s):  
Page Number ◽  
Linear Time ◽  
Dimensional Space ◽  
Time Algorithm ◽  
Linear Ordering ◽  
Linear Time Algorithm ◽  
The Sun ◽  
3 Dimensional ◽  
Book Embedding ◽  
Minimum Number

A book consists of a line in the 3-dimensional space, called the spine, and a number of pages, each a half-plane with the spine as boundary. A book embedding (p, r) of a graph consists of a linear ordering of p, of vertices, called the spine ordering, along the spine of a book and an assignment r, of edges to pages so that edges assigned to the same page can be drawn on that page without crossing. That is, we cannot find vertices u, v, x, y with p(u) < p(x) < p(v) < p(y), yet the edges uv and xy are assigned to the same page, that is r(uv) = r(xy). The book thickness or page number of a graph G is the minimum number of pages in required to embed G in a book. In this paper we consider the Sun Graph or the Trampoline graph and obtain the printing cycle for embedding the Sun Graph in a single page. We also give a linear time algorithm for such an embedding.

QUADRANGULAR REFINEMENTS OF CONVEX POLYGONS WITH AN APPLICATION TO FINITE-ELEMENT MESHES

2000 ◽  
Vol 10 (01) ◽  
pp. 1-40 ◽  
Author(s):  
MATTHIAS MÜLLER-HANNEMANN ◽  
KARSTEN WEIHE
Keyword(s):  
Finite Element ◽  
Convex Polygon ◽  
Linear Time ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Convex Polygons ◽  
Minimum Number ◽  
Three Dimensional Space

We present a linear–time algorithm that decomposes a convex polygon conformally into a minimum number of strictly convex quadrilaterals. Moreover, we characterize the polygons that can be decomposed without additional vertices inside the polygon, and we present a linear–time algorithm for such decompositions, too. As an application, we consider the problem of constructing a minimum conformal refinement of a mesh in the three–dimensional space, which approximates the surface of a workpiece. We prove that this problem is strongly [Formula: see text] –hard, and we present a linear-time algorithm with a constant approximation ratio of four.

scholarly journals Minimum Cell Connection in Line Segment Arrangements

2017 ◽  
Vol 27 (03) ◽  
pp. 159-176
Author(s):  
Helmut Alt ◽  
Sergio Cabello ◽  
Panos Giannopoulos ◽  
Christian Knauer
Keyword(s):  
Linear Time ◽  
Optimal Solution ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Constant Number ◽  
Line Segments ◽  
Straight Line ◽  
Minimum Number ◽  
Number Of Segments ◽  
Connection Problems

We study the complexity of the following cell connection problems in segment arrangements. Given a set of straight-line segments in the plane and two points [Formula: see text] and [Formula: see text] in different cells of the induced arrangement: [(i)] compute the minimum number of segments one needs to remove so that there is a path connecting [Formula: see text] to [Formula: see text] that does not intersect any of the remaining segments; [(ii)] compute the minimum number of segments one needs to remove so that the arrangement induced by the remaining segments has a single cell. We show that problems (i) and (ii) are NP-hard and discuss some special, tractable cases. Most notably, we provide a near-linear-time algorithm for a variant of problem (i) where the path connecting [Formula: see text] to [Formula: see text] must stay inside a given polygon [Formula: see text] with a constant number of holes, the segments are contained in [Formula: see text], and the endpoints of the segments are on the boundary of [Formula: see text]. The approach for this latter result uses homotopy of paths to group the segments into clusters with the property that either all segments in a cluster or none participate in an optimal solution.

Search Numbers in Networks with Special Topologies

2019 ◽  
Vol 19 (01) ◽  
pp. 1940004
Author(s):  
BOTING YANG ◽  
RUNTAO ZHANG ◽  
YI CAO ◽  
FARONG ZHONG
Keyword(s):  
Approximation Algorithms ◽  
Search Strategy ◽  
Linear Time ◽  
Time Algorithm ◽  
Time Transformation ◽  
Linear Time Algorithm ◽  
Optimal Layout ◽  
Explicit Formulas ◽  
Minimum Number ◽  
Search Numbers

In this paper, we consider the problem of finding the minimum number of searchers to sweep networks/graphs with special topological structures. Such a number is called the search number. We first study graphs, which contain only one cycle, and present a linear time algorithm to compute the vertex separation and the optimal layout of such graphs; by a linear-time transformation, we can find the search number of this kind of graphs in linear time. We also investigate graphs, in which every vertex lies on at most one cycle and each cycle contains at most three vertices of degree more than two, and we propose a linear time algorithm to compute their search number and optimal search strategy. We prove explicit formulas for the search number of the graphs obtained from complete k-ary trees by replacing vertices by cycles. We also present some results on approximation algorithms.

OPTIMUM GUARD COVERS AND m-WATCHMEN ROUTES FOR RESTRICTED POLYGONS

1993 ◽  
Vol 03 (01) ◽  
pp. 85-105 ◽  
Author(s):  
SVANTE CARLSSON ◽  
BENGT J. NILSSON ◽  
SIMEON NTAFOS
Keyword(s):  
Linear Time ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Np Hard ◽  
Art Gallery ◽  
Art Galleries ◽  
Art Gallery Problem ◽  
Natural Connection ◽  
Route Problem ◽  
Minimum Number

A watchman, in the terminology of art galleries, is a mobile guard. We consider several watchman and guard problems for different classes of polygons. We introduce the notion of vision spans along a path or route which provide a natural connection between the art gallery problem, the m-watchmen routes problem and the watchman route problem. We prove that finding the minimum number of vision points, i.e., static guards, along a shortest watchman route is NP-hard. We provide a linear time algorithm to compute the best set of static guards in a histogram polygon. The m-watchmen routes problem, minimize total length of routes for m watchmen, is NP-hard for simple polygons. We give a Θ(n3+n2m2)-time algorithm to compute the best set of m watchmen in a histogram.

Linear time algorithm for dominator chromatic number of trestled graphs

2019 ◽  
Vol 11 (06) ◽  
pp. 1950066
Author(s):  
S. Arumugam ◽  
K. Raja Chandrasekar
Keyword(s):  
Chromatic Number ◽  
Linear Time ◽  
Open Neighborhood ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Proper Coloring ◽  
Coloring Problem ◽  
Color Class ◽  
Minimum Number ◽  
Dominator Coloring

A dominator coloring (respectively, total dominator coloring) of a graph [Formula: see text] is a proper coloring [Formula: see text] of [Formula: see text] such that each closed neighborhood (respectively, open neighborhood) of every vertex of [Formula: see text] contains a color class of [Formula: see text] The minimum number of colors required for a dominator coloring (respectively, total dominator coloring) of [Formula: see text] is called the dominator chromatic number (respectively, total dominator chromatic number) of [Formula: see text] and is denoted by [Formula: see text] (respectively, [Formula: see text]). In this paper, we prove that the dominator coloring problem and the total dominator coloring problem are solvable in linear time for trestled graphs.

scholarly journals Prefix Block-Interchanges on Binary and Ternary Strings

2019 ◽  
Author(s):  
Md. Khaledur Rahman ◽  
M. Sohel Rahman
Keyword(s):  
Upper Bound ◽  
Genome Rearrangement ◽  
Linear Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Upper Bounds ◽  
Linear Time Algorithm ◽  
Minimum Number ◽  
Binary Strings ◽  
Better Than

AbstractThe genome rearrangement problem computes the minimum number of operations that are required to sort all elements of a permutation. A block-interchange operation exchanges two blocks of a permutation which are not necessarily adjacent and in a prefix block-interchange, one block is always the prefix of that permutation. In this paper, we focus on applying prefix block-interchanges on binary and ternary strings. We present upper bounds to group and sort a given binary/ternary string. We also provide upper bounds for a different version of the block-interchange operation which we refer to as the ‘restricted prefix block-interchange’. We observe that our obtained upper bound for restricted prefix block-interchange operations on binary strings is better than that of other genome rearrangement operations to group fully normalized binary strings. Consequently, we provide a linear-time algorithm to solve the problem of grouping binary normalized strings by restricted prefix block-interchanges. We also provide a polynomial time algorithm to group normalized ternary strings by prefix block-interchange operations. Finally, we provide a classification for ternary strings based on the required number of prefix block-interchange operations.

A LINEAR-TIME ALGORITHM FOR COVERING SIMPLE POLYGONS WITH SIMILAR RECTANGLES

1996 ◽  
Vol 06 (01) ◽  
pp. 79-102 ◽  
Author(s):  
REUVEN BAR-YEHUDA ◽  
EYAL BEN-HANOCH
Keyword(s):  
Image Compression ◽  
Linear Time ◽  
Simple Polygon ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Equivalent Problem ◽  
Orthogonal Polygon ◽  
Simple Polygons ◽  
Minimum Number ◽  
Incremental Update

We study the problem of covering a simple orthogonal polygon with a minimum number of (possibly overlapping) squares, all internal to the polygon. The problem has applications in VLSI mask generation, incremental update of raster displays, and image compression. We give a linear time algorithm for covering a simple polygon, specified by its vertices, with squares. Covering with similar rectangles (having a given x/y ratio) is an equivalent problem.

scholarly journals Total Roman {3}-Domination: The Complexity and Linear-Time Algorithm for Trees

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 293
Author(s):  
Xinyue Liu ◽  
Huiqin Jiang ◽  
Pu Wu ◽  
Zehui Shao
Keyword(s):  
Linear Time ◽  
Minimum Weight ◽  
Domination Number ◽  
Time Algorithm ◽  
Simple Graph ◽  
Linear Time Algorithm ◽  
Domination Problem ◽  
Np Complete ◽  
Chordal Bipartite Graphs ◽  
Dominating Function

For a simple graph G=(V,E) with no isolated vertices, a total Roman {3}-dominating function(TR3DF) on G is a function f:V(G)→{0,1,2,3} having the property that (i) ∑w∈N(v)f(w)≥3 if f(v)=0; (ii) ∑w∈N(v)f(w)≥2 if f(v)=1; and (iii) every vertex v with f(v)≠0 has a neighbor u with f(u)≠0 for every vertex v∈V(G). The weight of a TR3DF f is the sum f(V)=∑v∈V(G)f(v) and the minimum weight of a total Roman {3}-dominating function on G is called the total Roman {3}-domination number denoted by γt{R3}(G). In this paper, we show that the total Roman {3}-domination problem is NP-complete for planar graphs and chordal bipartite graphs. Finally, we present a linear-time algorithm to compute the value of γt{R3} for trees.

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