Connected Set Of Points Research Articles

Connected Point Set Domination of Fuzzy Graphs

Let  be a fuzzy Graph. Let  denote respectively the domination number, the point set domination number and the connected point set domination number of a fuzzy graph. A dominating set D of a fuzzy graph is said to be a point set dominating set of a fuzzy graph if for every  there exists a node  such that  is connected fuzzy graph. The minimum cardinality taken over all minimal point set dominating set is called the point set domination number of a fuzzy graph G and it is denoted by . A point set dominating set D of any fuzzy graph G is a connected point set dominating set if the fuzzy subgraph induced by D is connected. The minimum cardinality taken over all point set dominating set of a fuzzy graph G is said to be connected point set domination of a fuzzy graph G. It is denoted by . In this paper we concentrate on connected point set domination number of fuzzy graph and obtain some interesting results for new parameter of fuzzy graphs.

POINT VISIBILITY GRAPHS AND ${\mathcal O}$-CONVEX COVER

A visibility relation can be viewed as a graph: the uncountable graph of a visibility relationship between points in a polygon P is called the point visibility graph (PVG) of P. In this paper we explore the use of perfect graphs to characterize tractable subproblems of visibility problems. Our main result is a characterization of which polygons are guaranteed to have weakly triangulated PVGs, under a generalized notion of visibility called [Formula: see text]-visibility. Let [Formula: see text] denote a set of line orientations. A connected point set P is called [Formula: see text]-convex if the intersection of P with any line with orientation in [Formula: see text] is connected. Two points in a polygon are said to be [Formula: see text]-visible if there is an [Formula: see text]-convex path between them inside the polygon. Let [Formula: see text] denote the set of orientations perpendicular to orientations in [Formula: see text]. Let [Formula: see text] be the set of orientations θ such that a "reflex" local maximum in the boundary of P exists with respect to θ. Our characterization of which polygons have weakly-triangulated PVGs is based on restricting the cardinality and span of [Formula: see text]. This characterization allows us to exhibit a class of polygons admitting a polynomial algorithm for [Formula: see text]-convex cover.

Shape of an arbitrary finite point set in IR2

We study the shape of a finite point set in IR2, where the points are not bound to a regular grid like ℤ2. The shape of a connected point set in IR2 is captured by its boundary. For a finite point set the boundary is a directed graph that connects points identified as boundary points. We argue that to serve as a proper boundary definition the directed graph should regulate scale, be minimal, have an increasing interior and be consistent with the boundary definition of connected objects.

Sets Accessible at Each Point Only by Wild Arcs

A positional pathology as described in the title is shown to occur in three-space. We construct an arcwise accessible point set $M$ such that each arc to $M$ is locally knotted at uncountably many points. In addition we give examples of connected, locally connected point sets $S$ and $T$ which are accessible at each point only by wild arcs and tame arcs respectively. The point set $M$ has countably many components, and each of these is a tame finite $2$-complex. Moreover ${E^3} - M$ is locally connected and arcwise connected.

A lower bound for the critical probability in a certain percolation process

Consider a lattice L in the Cartesian plane consisting of all points (x, y) such that either x or y is an integer. Points with integer coordinates (positive, negative, or zero) are called vertices and the sides of the unit squares (including endpoints) are called links. Each link of L is assigned the designation active with probability p or passive with probability 1 − p, independently of all other links. To avoid trivial cases, we shall always assume 0 < p < 1. The lattice L, with the designations active or passive attached to the links, is called a random maze. A set of links is called connected if the points comprising the links (including endpoints) form a connected point set in the plane.

A connected point set in the plane which spirals down on each of its points

A connected point set in the plane which spirals down on each of its points

On decomposition of continua into aposyndetic continua

Introduction. This paper generalizes certain methods of decomposition of compact metric continua due to R. L. Moore [3; 4] and G. T. Whyburn [9; 10; 13]. While their methods yield acyclic continuous curves, hyperspaces are obtained here which are aposyndetic [1 ] continua. The concept of a con. tinuum being aposyndetic is a generalization of the concept of continuous curves and was introduced in 1941 by F. B. Jones. In compact metric continua, this idea is equivalent to Whyburn's notion of semi-locally-connectedness [14; 2]. A continuum M, i.e., a closed and connected point set, is said to be aposyndetic at a point p with respect to a point x provided that there exists a subcontinuum N of M and an open subset 0 of M such that M-xDNDODp. If M is aposyndetic at a point p with respect to each point x of M-p, then M is said to be aposyndetic at p. It is said that M is aposyndetic if M is aposyndetic at each of its points. In an early paper [10], Whyburn made use of connected cuttings of a compact metric continuum M to obtain a decomposition of M into an acyclic continuous curve. Later, he made use of nonseparated cuttings [13] to obtain a decomposition of a continuous curve into a nondegenerate acyclic continuous curve. Certain of these theorems concerning nonseparated cuttings are generalized in obtaining an aposyndetic decomposition. Moore [3] has obtained decomposition theorems by use of certain sets M(P) defined as follows: For each point P of a compact continuum M, let M(P) denote the set of all points X of M such that there do not exist uncountably many different points each separating P from X in M. He proved the following theorem: If M is a compact metric continuum and G is the collection of all point sets M(P) for all points P of M, then G is an upper semi-continuous collection of disjoint continua filling up M and G is an acyclic continuous curve with respect to its elements as points. The definition of the sets M(P) due to Moore may be generalized in the following manner: Suppose that M is a continuum and that p is a point of M.

Non-Separated Cuttings of Connected Point Sets

Non-Separated Cuttings of Connected Point Sets

Non-isolated critical points of functions

The isolated critical points of real functions of n independent real variables have been treated in an elegant manner by Marston Morse.J This treatment obtains definite relations between the numbers of critical points of w + 1 types that appear in a bounded portion of the space of the independent variables. Morse requires his functions to have continuous third partial derivatives in the neighborhoods of the critical points and imposes conditions that are sufficient to insure the existence of at most a finite number of such points in the domain under consideration. In the present note§ we consider functions that have continuous first partial derivatives and may have an infinite number of critical points, or even continua of such points, in the given domain. As a special case of our results we obtain the minimax principle of Birkhoff in the modified form given by Bieberbach.|| Let 5 denote the space of the n real variables x\> x%, • • • , xn, and let (x) = (xi, x2, • • • , xn) denote a point in this space. Let R be a bounded,^ open, and connected point set in S and let C, the boundary of R, be connected. Let the real function f(xi, • • • , xn) be single-valued and continuous

Concerning collections of cuttings of connected point sets

Two point sets are mutually separated if they are mutually exclusive and neither of them contains a limit point of the other. A point set M is said to be not connected or connected according as M is or is not the sum of two non-vacuous mutually separated point sets. A subset X of a connected point set M is said to be a cutting of M, or is said to cut M, provided the set of points M—X is not connected; X is called an irreducible cutting of M provided X cuts M but no proper subset of X cuts M. If A and B are subsets of a connected point set Af, the subset X of M is said to separate A and B in My or to cut M between A and B, provided that M~X is the sum of two mutually separated sets Ma(X) and Mb(X) containing A and B respectively. In this paper the ordinary notation of point set theory will be employed, for example, X = X + X ' , whereX is the set of all limit points of the set X, K c H means that the set K is a subset of the set H, K H means the set of points common to K and Hy F(R) denotes the boundary of the set i?, etc. In addition, if X is a cutting of a connected point set M then, unless otherwise stated, the equation M—X — Ma(X) + Mb(X) is to be interpreted as meaning that M—X is the sum of two mutually separated sets Ma(X) and Mb(X) which contain the sets A and B respectively. 1. Cuttings of Connected Sets in General. The point sets considered in this section are assumed to lie in a separable metric space which will be denoted by 5. It is apparent from the proofs of some of the theorems, however, that they hold in even more general space.

A theorem on connected point sets

A theorem on connected point sets

A theorem on connected point sets which are connected im kleinen

A theorem on connected point sets which are connected im kleinen

Certain Theorems Relating to Plane Connected Point Sets

Certain Theorems Relating to Plane Connected Point Sets

A theorem concerning connected point sets

A theorem concerning connected point sets

Certain theorems relating to plane connected point sets

Certain theorems relating to plane connected point sets

The converse of the theorem concerning the division of a plane by an open curve

In his paper, On the foutndations of plane analysis situs,t R. L. Moore proved that if 1 is an open curvet and S is the set of all points, then S 1 = Si + S2, where Si and S2 are connected point sets such that every arc from a point of Si to a point of S2 contains at least one point of 1.? Clearly the sets Si and S2 are non-compact.11 Professor Moore's theorem is proved on the basis of his set of axioms 13. Thus the theorem is true in certain spaces which are neither metrical, descriptive, nor separable. This theorem for open curves is analogous to the theorem of Jordan,? that a simple closed curve lying wholly within a plane decomposes the plane into an inside and an outside region. The converse of this theorem for simple closed curves was first formulated by Schoenflies,** who makes use of metrical properties in his proof. A different proof has been given by Lennes,tt who uses straight lines. R. L. Moore has pointed out that, on the basis of 13, an argument similar in large part to that of Lennes can be carried through with the use of arcs and closed curves. The object of the present paper is to show that the converse of the open curve theorem holds in spaces satisfying 13. The statement of the converse theorem is as follows: