Redundantly globally rigid braced triangulations

Sufficient conditions for the global rigidity of graphs

Sensor and network topologies of rigid formations with distance information between mobile autonomous agents are considered. An approach based on rigidity for creating such topologies was suggested in our previous work. Here, we first illustrate some potential scenarios on formations that require unambiguity in the knowledge of distances between every pair of agents in a formation. Then, we show how a stronger type of rigidity, namely global rigidity, plays a role in creating such unambiguous formations. We draw out and summarize some relevant results from the related mathematical theory of global rigidity; and present some new results on globally rigid formations.

Firedmann model Φ of positive space curvature, vanishing pressure and cosmological constant when isometrically imbedded as a hypersurface in five dimensional Minkowski spaceM 5 is globally rigid: if F(Φ) and F′(Φ) are isometric imbeddings inM 5 there is a motion π of M5 such that F= π°f. The big bang singularity is the vertex of a null half-cone inM 5. Global rigidity leads to an invariant characterization of the singularity. The structure of matter at the singularity is governed by the de Sitter group.

We examine the generic local and global rigidity of various graphs in ℝd . Bruce Hendrickson showed that some necessary conditions for generic global rigidity are (d+1)-connectedness and generic redundant rigidity, and hypothesized that they were sufficient in all dimensions. We analyze two classes of graphs that satisfy Hendrickson’s conditions for generic global rigidity, yet fail to be generically globally rigid. We find a large family of bipartite graphs for d>3, and we define a construction that generates infinitely many graphs in ℝ5. Finally, we state some conjectures for further exploration.

The graph theoretic facility layout problem (GTFLP) seeks the best spatial arrangement of a set of facilities, which minimizes the total transportation cost of material flow between facilities, or maximizes total facility adjacency benefits. The GTFLP has applications in arranging rooms within a building floorplan, placing machines on a factory floor, controls on an instrument panel, or components on a circuit board. For this reason, the GTFLP is an important problem within industrial design. The GTFLP comprises two phases: the generation of a highly weighted maximal planar graph (MPG), whose edges specify adjacencies required in the layout, and the drawing of an orthogonal geometric dual to this MPG, called the block plan, under given area requirements for each facility. The first phase is NP-complete, and has been researched extensively; in the absence of a polynomial time algorithm, it appears there is little future for further research in this area. The second phase however has received far less attention due to the difficulties in not only determining algorithms for generating layouts, but also the hurdles encountered in automating these methods. This paper presents a new approach to finding a block plan from an arbitrary MPG, with facility area requirements.

A maximal planar graph is a simple planar graph in which every face is a triangle. We show here that such graphs with maximum degree Δ and diameter two have no more than 3/2Δ + 1 vertices. We also show that there exist maximal planar graphs with diameter two and exactly [3/2Δ + 1] vertices.

A graph G on n vertices is called non-universal if its maximum degree is at most $$n-2$$ . In this paper, we give a structural characterization for non-universal maximal planar graphs with diameter two. In precise, we find 10 basic graphs, and then generate all 25 non-universal maximal planar graphs with diameter two by adding repeatedly and appropriately 3-vertices to some of these 10 basic graphs. As an application, we show that maximal planar graphs with diameter two are pancyclic except five special graphs.

The $(\Delta,D)$ (degree/diameter) problem consists of finding the largest possible number of vertices $n$ among all the graphs with maximum degree $\Delta$ and diameter $D$. We consider the $(\Delta,D)$ problem for maximal planar bipartite graphs, that is, simple planar graphs in which every face is a quadrangle. We obtain that for the $(\Delta,2)$ problem, the number of vertices is $n=\Delta+2$; and for the $(\Delta,3)$ problem, $n= 3\Delta-1$ if $\Delta$ is odd and $n= 3\Delta-2$ if $\Delta$ is even. Then, we prove that, for the general case of the $(\Delta,D)$ problem, an upper bound on $n$ is approximately $3(2D+1)(\Delta-2)^{\lfloor D/2\rfloor}$, and another one is $C(\Delta-2)^{\lfloor D/2\rfloor}$ if $\Delta\geq D$ and $C$ is a sufficiently large constant. Our upper bounds improve for our kind of graphs the one given by Fellows, Hell and Seyffarth for general planar graphs. We also give a lower bound on $n$ for maximal planar bipartite graphs, which is approximately $(\Delta-2)^{k}$ if $D=2k$, and $3(\Delta-3)^k$ if $D=2k+1$, for $\Delta$ and $D$ sufficiently large in both cases.

On The Number of Subgraphs of Prescribed Type of Planar Graphs With A Given Number of Vertices

In this paper, we investigate the problem of graph list colouring in the on-line setting. We provide several results on paintability of graphs in the model introduced by Schauz [18] and Zhu [25]. We prove that the on-line version of Ohba’s conjecture is true for the class of planar graphs. We show that the conjecture for partial list colouring on-line holds for several graph classes, namely claw-free graphs, maximal planar graphs, series-parallel graphs, and chordal graphs.

A subset of vertics W and a planer graph G is a face-independent vertex cover (FIVC) if there exists a plane embedding of G such that each boundary contains exactly one vertex from W. In general, verifying if a plane graph admits a VIFC is an NP-complete problem. If G is a maximal planar graph, then the existence of a FIVC in G can be efficiently tested by using the so-called t-paths between odd-degree vertics of G (a t-path is a sequence of adjacent triangles). Also for Halin Graphs (i.e., skirted trees), its FIVC’s can be generated in polynomial time.

Vertex splitting replaces a vertex by two copies and partitions its incident edges amongst the copies. This problem has been studied as a graph editing operation to achieve desired properties with as few splits as possible, most often planarity, for which the problem is $$\textsf{NP}$$ -hard. Here we study how to minimize the number of splits to turn a plane graph into an outerplane one. We tackle this problem by establishing a direct connection between splitting a plane graph to outerplanarity, finding a connected face cover, and finding a feedback vertex set in its dual. We prove $$\textsf{NP}$$ -completeness for plane biconnected graphs, while we show that a polynomial-time algorithm exists for maximal planar graphs. Finally, we provide upper and lower bounds for certain families of maximal planar graphs.

SUMMARYAfter listing some general formulae for sampling in n‐dimensional space, the author considers the one‐dimensional case: the estimation of the length of a line segment by counting the number of points that happen to fall within the segment. If the points are equidistantly located, the variance of the estimate is a strictly periodic function of the length of the segment. This systematic sample has a higher efficiency than simple and stratified random samples of the same intensity.With some modifications, the results carry over to the two‐dimensional case: the estimation of the area of a plane figure by counting the number of sample points falling inside the figure. However, the strict periodicity of the variance in the one‐dimensional systematic case is replaced by a ‘Zitterbewegung’. The magnitude of this oscillation is seen to be very different for figures of different shapes.Some results are presented also for the estimation of areas by line transects, and for the estimation of volumes by aid of lattices of points in R3, and R4. Some comments are also given on the practical implications of the results for sampling in the plane.

Maximal planar graphs of inscribable type and diagonal flips

The Kempe change is the essence of the Kempe’s “proof” of the Four Color Conjecture (Kempe, Am. J. Math. 2(3), 193–200 (1879)), by which a new 4-coloring of a maximal planar graph can be generated from a given 4-coloring. The fundamental reason why it fails to prove the Four Color Conjecture by using this technique is that there exist many maximal planar graphs G such that the set of all 4-colorings of G can not be generated by applying Kempe change from any given 4-colorings of G. Nevertheless, due to the NP-completeness of the vertex coloring of graphs, Kempe change has been a fundamental and most powerful tool in the study of theory, algorithm, and application of graph colorings since 1879. This chapter is devoted to the introduction of related theory on Kempe change, including Kempe equivalence of colorings, $$\sigma $$ σ -characteristic graphs (which depicts the relation of all colorings), and non-Kempe graphs.