Finding minimal convex nested polygons

There are many practical applications that require simplification of polylines. Some of the goals are to reduce the amount of information necessary to store, improve processing time, or simplify editing. The simplification is usually done by removing some of the vertices, making the resultant polyline go through a subset of the source polyline vertices. However, such approaches do not necessarily produce a new polyline with the minimum number of vertices. The approximate solution to find a polyline, within a specified tolerance, with the minimum number of vertices is described in this paper.

For a graph G(V, E) with order n ≥ 2, the locating code of a vertex v is a finite vector representing distances of v with respect to vertices of some ordered subset W of V(G). The set W is a locating set of G(V, E) if distinct vertices have distinct codes. A locating set containing a minimum number of vertices is a minimum locating set for G(V, E). The locating number denoted by loc (G) is the number of vertices in the minimum locating set. Let R be a commutative ring with identity 1 ≠ 0, the zero-divisor graph denoted by Γ(R), is the (undirected) graph whose vertices are the nonzero zero-divisors of R with two distinct vertices joined by an edge when the product of vertices is zero. We introduce and investigate locating numbers in zero-divisor graphs of a commutative ring R. We then extend our definition to study and characterize the locating numbers of an ideal based zero-divisor graph of a commutative ring R.

This paper discusses the receiver design of a destination node supporting multiple antennas for an alternate-relaying decode-and-forward (DF) cooperative communication system. We exploit the structure of the received signal to develop an optimal data detection algorithm at the destination. It is shown that the optimal algorithm can be implemented by parallel detectors, each based on a family of Bahl, Cocke, Jelinek and Raviv (BCJR) algorithms. The proposed optimal algorithm requires to receive and store all packets before performing data detection. To avoid this, a sub-optimal algorithm is also proposed. Unlike the optimal algorithm, the sub-optimal one exploits two consecutive received packets to detect one packet. It turns out that the sub-optimal algorithm has less reduced delay, complexity, memory size and bandwidth loss with a slight increase of the bit-error-rate. The detection performance of the proposed algorithms is evaluated via Monte Carlo simulations, and the results demonstrate their effectiveness.

Constructions of bi-regular cages

On the order of [formula omitted]-cages of even girth

The problem of cutting a convex polygon P out of a piece of paper Q with minimum total cutting length is a well studied problem in computational geometry. Researchers studied several variations of the problem, such as P and Q are convex or non-convex polygons and the cuts are line cuts or rays cuts. In this paper we consider yet another variation of the problem where Q is a circle and P is convex polygon such that P is bounded by a half circle of Q and all the cuts are line cuts. We give a simple linear time O(log n)-approximation algorithm for this problem where n is the number of vertices of P.

Given a sequence of ordered convex polygons in which the adjacent polygons may intersect with each other, but the nonadjacent polygons do not intersect, a start point s, and an end point t in the plane, our goal is to obtain a shortest path that starts from s, visits each given polygon in order, and ends at t finally. We converted the touring polygons problem into the problem of computing the shortest path of visiting the disjoint line segments by analyzing the geometrical features of the given convex polygons, and preprocessing the intersection points of the jointed polygons, and using a forward partition process combined with a backward search process for finding the access edge of each convex polygon. Thus, we proposed an 0(max{n, klog⁁2k}) algorithm for solving the original problem, where n is the total number of vertices of the given polygons and k is the total number of polygons.

Algorithms for line clipping against convex polygon have been studied for a long time and many research papers have been published so far. In spite of the latest graphical hardware development and significant increase of performance the clipping is still a bottleneck of the graphical pipeline. This paper presents a new robust and fast algorithm for line clipping by a convex polygon. The algorithm uses a small preprocessing in order to obtain significant speed up. The proposed algorithm is especially convenient for applications where points or lines are represented in homogeneous coordinates. The algorithm does not use division in floating point representation as the resulting points are in homogeneous coordinates. The algorithms benefit if vector-vector hardware supported operations can be used

Fast computation of the modality of polygons

Let G=(V, E) be a graph. Let D be a minimum total dominating set of G. If V–D contains a total dominating set D’ of G, then D’ is called an inverse total dominating set with respect to D. The inverse total domination number of G is the minimum number of vertices in an inverse total dominating set of G. We initiate the study of inverse total domination in graphs and present some bounds and some exact values for . Also, some relationships between and other domination parameters are established.

Polyhedral Manifolds With Few Vertices

LetG=(V,E)be a connected simple graph and letMbe a nonempty subset ofV. TheM-distance pattern of a vertexuinGis the set of all distances fromuto the vertices inM. If the distance patterns of all vertices inVare distinct, then the setMis a distance pattern distinguishing set ofG. A graphGwith a distance pattern distinguishing set is called a distance pattern distinguishing graph. Minimum number of vertices in a distance pattern distinguishing set is called distance pattern distinguishing number of a graph. This paper initiates a study on the problem of finding distance pattern distinguishing number of a graph and gives bounds for distance pattern distinguishing number. Further, this paper provides an algorithm to determine whether a graph is a distance pattern distinguishing graph or not and hence to determine the distance pattern distinguishing number of that graph.

The distinguishing number D(G) of a graph G is the least integer d such that G has a vertex labeling with d labels that is preserved only by a trivial automorphism. The distinguishing stability of a graph G is denoted by and is the minimum number of vertices whose removal changes the distinguishing number. We show that for every natural number k, there exists a graph G such that . Then we give some upper bounds for . We also study the edge distinguishing stability number (distinguishing bondage number) of G.

A homomorphism from an oriented graph $G$ to an oriented graph $H$ is a mapping $\varphi$ from the set of vertices of $G$ to the set of vertices of $H$ such that $\overrightarrow{\varphi(u)\varphi(v)}$ is an arc in $H$ whenever $\overrightarrow{uv}$ is an arc in $G$. The oriented chromatic index of an oriented graph $G$ is the minimum number of vertices in an oriented graph $H$ such that there exists a homomorphism from the line digraph $LD(G)$ of $G$ to $H$ (Recall that $LD(G)$ is given by $V(LD(G))=A(G)$ and $ \overrightarrow{ab}\in A(LD(G))$ whenever $a=\overrightarrow{uv}$ and $b=\overrightarrow{vw}$). We prove that every oriented subcubic graph has oriented chromatic index at most $7$ and construct a subcubic graph with oriented chromatic index $6$.

The strong matching preclusion number of a graph is the minimum number of vertices and edges whose deletion results in a graph that has neither perfect matchings nor almost-perfect matchings. This is an extension of the matching preclusion problem and has recently been introduced by Park and Ihm.15In this paper, we examine properties of strong matching preclusion for alternating group graphs, by finding their strong matching preclusion numbers and categorizing all optimal solutions. More importantly, we prove a general result on taking a Cartesian product of a graph with K2(an edge) to obtain the corresponding results for split-stars.