Rectangle Graphs Research Articles

Secure connected domination and secure total domination in unit disk graphs and rectangle graphs

Given a graph G with vertex set V, a secure connected (resp. total) dominating set of G is a connected (resp. total) dominating set S⊆V with the property that for each u∈V∖S, there exists v∈S adjacent to u such that (S∪{u})∖{v} is a connected (resp. total) dominating set of G. The minimum secure connected dominating set (or, for short, MSCDS) (resp. minimum secure total dominating set (or, for short, MSTDS)) problem is to find an MSCDS (resp. MSTDS) in a given graph.In this paper, we initiate to consider complexity and algorithmic aspects of the MSCDS problem and the MSTDS problem in unit disk graphs and rectangle graphs. Firstly, we show that the decision version of the MSCDS problem is NP-complete in unit square graphs and unit disk graphs. Then we show that the decision version of the MSTDS problem is NP-complete even in grid graphs (a subclass of unit square graphs and unit disk graphs). Secondly, we give linear-time constant-approximation algorithms for the two problems in unit square graphs, unit disk graphs and unit-height rectangle graphs. Thirdly, we propose a PTAS for the MSTDS problem in unit square graphs and unit disk graphs. Finally, we show that the two problems in proper rectangle graphs are APX-hard. Further we give an explicit lower bound 1.00147 on efficient approximability for the two problems in proper rectangle graphs unless P=NP.

Partial Latin rectangle graphs and autoparatopism groups of partial Latin rectangles with trivial autotopism groups

An r×s partial Latin rectangle (lij) is an r×s matrix containing elements of {1,2,…,n}∪{⋅} such that each row and each column contain at most one copy of any symbol in {1,2,…,n}. An entry is a triple (i,j,lij) with lij≠⋅. Partial Latin rectangles are operated on by permuting the rows, columns, and symbols, and by uniformly permuting the coordinates of the set of entries. The stabilizers under these operations are called the autotopism group and the autoparatopism group, respectively.We develop the theory of symmetries of partial Latin rectangles, introducing the concept of a partial Latin rectangle graph. We give constructions of m-entry partial Latin rectangles with trivial autotopism groups for all possible autoparatopism groups (up to isomorphism) when: (a) r=s=n, i.e., partial Latin squares, (b) r=2 and s=n, and (c) r=2 and s≠n.

Self-scaled bounds for atomic cone ranks: applications to nonnegative rank and cp-rank

The nonnegative rank of a matrix A is the smallest integer r such that A can be written as the sum of r rank-one nonnegative matrices. The nonnegative rank has received a lot of attention recently due to its application in optimization, probability and communication complexity. In this paper we study a class of atomic rank functions defined on a convex cone which generalize several notions of "positive" ranks such as nonnegative rank or cp-rank (for completely positive matrices). The main contribution of the paper is a new method to obtain lower bounds for such ranks which improve on previously known bounds. Additionally the bounds we propose can be computed by semidefinite programming. The idea of the lower bound relies on an atomic norm approach where the atoms are self-scaled according to the vector (or matrix, in the case of nonnegative rank) of interest. This results in a lower bound that is invariant under scaling and that is at least as good as other existing norm-based bounds. We mainly focus our attention on the two important cases of nonnegative rank and cp-rank where our bounds satisfying interesting properties: For the nonnegative rank we show that our lower bound can be interpreted as a non-combinatorial version of the fractional rectangle cover number, while the sum-of-squares relaxation is closely related to the Lov\'asz theta number of the rectangle graph of the matrix. We also prove that the lower bound inherits many of the structural properties satisfied by the nonnegative rank such as invariance under diagonal scaling, subadditivity, etc. We also apply our method to obtain lower bounds on the cp-rank for completely positive matrices. In this case we prove that our lower bound is always greater than or equal the plain rank lower bound, and we show that it has interesting connections with combinatorial lower bounds based on edge-clique cover number.

Mutual witness proximity graphs

This paper describes one variation on witness proximity graphs called mutual witness proximity graphs. Two witness proximity graphs are said to be mutual when, given two sets of points A and B, A is the vertex set of the first graph and the witness set of the second one, while B is the witness set of the first graph and the vertex set of the second one. We show that in the union of two mutual witness Delaunay graphs, there are always at least ⌈n−22⌉ edges, where n=|A|+|B|, which is tight in the worst case. We also show that if two mutual witness Delaunay graphs are complete, then the sets A and B are circularly separable; if two mutual witness Gabriel graphs are complete, then the sets A and B are linearly separable; but two mutual witness rectangle graphs might be complete, with A and B not linearly separable.

Witness Rectangle Graphs

In a witness rectangle graph (WRG) on vertex point set P with respect to witness point set W in the plane, two points x, y in P are adjacent whenever the open isothetic rectangle with x and y as opposite corners contains at least one point in W. WRGs are representative of a larger family of witness proximity graphs introduced in two previous papers. We study graph-theoretic properties of WRGs. We prove that any WRG has at most two non-trivial connected components. We bound the diameter of the non-trivial connected components of a WRG in both the one-component and two-component cases. In the latter case, we prove that a graph is representable as a WRG if and only if each component is a connected co-interval graph, thereby providing a complete characterization of WRGs of this type. We also completely characterize trees drawable as WRGs. In addition, we prove that a WRG with no isolated vertices has domination number at most four. Moreover, we show that any combinatorial graph can be drawn as a WRG using a combination of positive and negative witnesses. Finally, we conclude with some related results on the number of points required to stab all the rectangles defined by a set of n points.

구글 지도에 통계정보를 표현하기 위한 R 함수 개발

구글 지도는 지리 정보를 갖는 데이터에 대한 통계정보를 제공하기 위한 보편화된 수단의 하나로 자리매김하고 있다. 본 연구에서는 R에서 구글 지도를 활용하는 방법에 대해 소개하고, 구글 지도상에 다양한 통계그래프를 표현하기 위한 R 함수를 개발한다. 개발된 함수를 통하여 막대그래프, 원형그래프, 사각형그래프 등과 같은 다양한 통계그래프를 지도상에 표현할 수 있다. Google map has become one of the most recognized and comfortable means for providing statistical information of geographically referenced data. In this article, we introduce R functions to embed google map images on R interface and develop a function to represent statistical graphs such as bar graph, pie chart, and rectangle graph on a google map images.

On cartesian product of vertex measurable graphs

In [2] for any graph G, a σ algebra ℑ of vertex induced sub graphs of G, is defined and many of its properties are studied. Let G 1 and G 2 be two simple graphs. Let (G 1, ℑ 1) and (G 2, ℑ 2) be two vertex measure spaces. In this paper we introduce a σ algebra ℑ 1 × ℑ 2, which consists of all vertex induced sub graphs of G 1 × G 2, and it contains every vertex measurable rectangle graph of the form H 1 × H 2, H 1 ∈ ℑ1 and H 2 ∈ ℑ 2. Also, we introduce an elementary vertex measurable rectangle graph and a vertex graph monotone class and study some of their properties. Finally, we prove ℑ 1 × ℑ 2 is the smallest vertex graph monotone class which contains all elementary vertex measurable rectangle graphs.

Minimum vertex cover in rectangle graphs

We consider the Minimum Vertex Cover problem in intersection graphs of axis-parallel rectangles on the plane. We present two algorithms: The first is an EPTAS for non-crossing rectangle families, rectangle families R where R 1 ∖ R 2 is connected for every pair of rectangles R 1 , R 2 ∈ R . This algorithm extends to intersection graphs of pseudo-disks. The second algorithm achieves a factor of ( 1.5 + ε ) in general rectangle families, for any fixed ε > 0 , and works also for the weighted variant of the problem. Both algorithms exploit the plane properties of axis-parallel rectangles in a novel way.

Hamiltonian circuits in interval graph generalizations

We consider the problem of finding a Hamiltonian circuit in some classes of intersection graphs which generalize interval graphs. We show that the problem is NP-complete for undirected path graphs (and hence chordal graphs), double interval graphs, and rectangle graphs.

Stacks and splits of partially ordered sets

The dimension of a partially ordered set ( X, P) is the smallest positive integer t for which there exists a function ƒ which assigns to each x∈ X a sequence {ƒ(x)(i): 1⩽i⩽t} of real numbers so that xles; y in P if and only if ƒ(x)(i)⩽ƒ(y)(i) for each i=1,2,…, t. The interval dimension of ( X, P) is the smallest integer t for which there exists a function F which assigns to each x∈ X a sequence { F( x)( i): 1⩽ i⩽ t} of closed intervals of the real line R so that x < y in P if and only if a < b in R for every a∈ F( x)( i), b∈ F( y)( i), and i = 1,2…, t. For t⩾2, a partially ordered set (poset) is said to be t-irreducible (resp. t-interval irreducible) if it has dimension t (resp. interval dimension t), and every proper subposet has dimension (resp. interval dimension) less than t. The only 2-irreducible poset is a two element anti-chain, and the only 2-interval irreducible poset is the free sum of two chains each having two points. In sharp contrast, the collection R of all 3-irreducible posets consists of 9 infinite families and 18 odd examples, and the collection R I of all 3-interval irreducible posets is sufficiently complex to have avoided complete determination as of this date. Trotter and Moore determined R from Gallai's forbidden subgraph characterization of comparability graphs. David Kelly independently determined R by a lattice theoretic argument combined with the characterization of planar lattices Kelly and Ivan Rival had previously obtained. In this paper, we introduce a new operation called a stack which we will apply to posets of height one. In some ways the stack operation is an inverse of the split operation on posets previously defined by Kimble. These operations behave predictably with respect to dimension and interval dimension. In particular, the stack of a poset of height one plays a role in interval dimension theory which is analogous to the role played by the completion by cuts in dimension theory. As a consequence, we can exploit the similarities to Kelly's approach to the determination of R to produce a relatively compact argument to determine the collection R ( I, 1) of all 3-interval irreducible posets of height one. This characterization problem has immediate combinatorial connections with a wide range of well-known forbidden subgraph problems including interval graphs, rectangle graphs, and circular arc graphs.

A characterization of robert's inequality for boxicity

F.S. Roberts defined the boxicity of a graph G as the smallest positive integer n for which there exists a function F assigning to each vertex xϵG a sequence F(x)(1),F(x)(2),…, F(x)(n) of closed intervals of R so that distinct vertices x and y are adjacent in G if and only if F(x)(i)∩F(y)(i)≠∅ fori = 1, 2, 3, …, n. Roberts then proved that if G is a graph having 2n + 1 vertices, thentheboxicityofGisatmostn. In this paper, we provide an explicit characterization of this inequality by determining for each n ⩾ 1 the minimum collection Cn of graphs so that a graph G having 2n + 1 vertices has boxicity n if and only if it contains a graph from Cn as an induced subgraph. We also discuss combinatorial connections with analogous characterization problems for rectangle graphs, circular arc graphs, and partially ordered sets.