Search

A Roman dominating function (RDF) on a graph G is a function f : V (G) → {0, 1, 2} satisfying the condition that every vertex u with f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. The weight of an RDF f is the sum of the weights of the vertices under f. The Roman domination number, γR(G) of G is the minimum weight of an RDF in G. The Roman domination polynomial of a graph G of order n is the polynomial RD(G, x) = Σ2ni=γR(G) dR(G, i)xi, where dR(G, i) is the number of RDFs of G with weight i. In this paper we prove properties of Roman domination polynomials and determine RD(G, x) in several classes of graphs G by new approaches. We also present bounds on the number of all Roman domination polynomials in a graph. [ABSTRACT FROM AUTHOR]

A signed graph is a graph with edges marked positive and negative; it is unbalanced if some cycle has negative sign product. We introduce the concept of vector valued switching function in signed graphs, which extends the concept of switching to higher dimensions. Using this concept, we define balancing dimension and strong balancing dimension for a signed graph, which can be used for a new classification of degree of imbalance of unbalanced signed graphs. We provide bounds for the balancing and strong balancing dimensions, and calculate these dimensions for some classes of signed graphs. [ABSTRACT FROM AUTHOR]

This paper introduces certain relations between 1-edge contraction and the total vertex stress and the confluence number of a graph. A main result states that if a graph G with ζ(G) = k ≥ 2 has an edge vivj and a ζ-set CG such that vi, vj ∈ CG then, ζ(G/vivj ) = k − 1. In general, either S(G/ei) ≤ S(G/ej ) or S(G/ej ) ≤ S(G/ei) is true. This observation leads to an investigation into the question: for which edge(s) ei will S(G/ei) = max{S(G/ej ) : ej ∈ E(G)} and for which edge(s) will S(G/ej ) = min{S(G/el) : el∈ E(G)}? [ABSTRACT FROM AUTHOR]

Let G = (V, E) be a simple connected graph. A perfect matching (or Kekulé structure in chemical literature) of G is a set of disjoint edges which covers all vertices of G. The anti-forcing number of G is the smallest number of edges such that the remaining graph obtained by deleting these edges has a unique perfect matching and is denoted by af(G). For every m ∈ N, the mth power of G, denoted by Gm, is a graph with the same vertex set as G such that two vertices are adjacent in Gm if and only if their distance is at most m in G. In this paper, we study the anti-forcing number of the powers of some graphs. [ABSTRACT FROM AUTHOR]

A new concept known as the orthogonal component graph associated with a finite-dimensional vector space over a finite field has been recently added as another class of algebraic graphs. In these types of graphs, the vertices will be all the possible non-zero linear combinations of orthogonal basis vectors, and any two vertices will be adjacent if the corresponding vectors are orthogonal. In this paper, we discuss the various colorings and structural properties of orthogonal component graphs. [ABSTRACT FROM AUTHOR]

For a graph G, we define a total k-labeling ϕ as a combination of an edge labeling ϕe : E(G) → {1, 2, . . ., ke} and a vertex labeling ϕv : V (G) → {0, 2, . . ., 2kv}, where k = max {ke, 2kv}. The total k-labeling ϕ is called a vertex irregular reflexive k-labeling of G if any pair of vertices u, u' have distinct vertex weights wtϕ(u) ≠ wtϕ(u'), where wtϕ(u) = ϕ(u) + Σuu'∈E(G) ϕ(uu' ) for any vertex u ∈ V (G). The smallest value of k for which such a labeling exists is called the reflexive vertex strength of G, denoted by rvs(G). In this paper, we present a new lower bound for the reflexive vertex strength of any graph. We investigate the exact values of the reflexive vertex strength of generalized friendship graphs, corona product of two paths, and corona product of a cycle with isolated vertices by referring to the lower bound. This study discovers some interesting open problems that are worth further exploration. [ABSTRACT FROM AUTHOR]

The coalition in a graph G consists of two disjoint sets of vertices V1 and V2, neither of which is a dominating set but whose union V1 ∪ V2, is a dominating set. A coalition partition in a graph G is a vertex partition π = {V1, V2, . . ., Vk} such that every set Vi ∈ π is not a dominating set but forms a coalition with another set Vj ∈ π which is not a dominating set. The coalition number C(G) equals the maximum k of a coalition partition of G. In this paper, we compute the coalition numbers of all cubic graphs of order at most 10. [ABSTRACT FROM AUTHOR]

The PI index of a graph G is given by PI(G) = Σe∈E(G) (|V (G)|−NG(e)), where NG(e) is the number of equidistant vertices for the edge e. Various topological indices of bicyclic graphs have already been calculated. In this paper, we obtained the exact value of the PI index of bicyclic graphs. We also explore the extremal graphs among all bicyclic graphs with respect to the PI index. Furthermore, we calculate the PI index of a cactus graph and determine the extremal values of the PI index among cactus graphs. [ABSTRACT FROM AUTHOR]

A graphoidal cover of a graph G (not necessarily finite) is a collection ψ of paths (not necessarily finite, not necessarily open) satisfying the following axioms: (GC-1) Every vertex of G is an internal vertex of at most one path in ψ, and (GC-2) every edge of G is in exactly one path in ψ. The pair (G, ψ) is called a graphoidally covered graph and the paths in ψ are called the ψ-edges of G. In a graphoidally covered graph (G, ψ), two distinct vertices u and v are ψ-adjacent if they are the ends of an open ψ-edge. A graphoidally covered graph (G, ψ) in which no two distinct vertices are ψ-adjacent is called ψ-independent and the graphoidal cover ψ is called a totally disconnecting graphoidal cover of G. Further, a graph possessing a totally disconnecting graphoidal cover is called a graphoidally independent graph. The aim of this paper is to establish complete characterization of graphoidally independent infinite cactus. [ABSTRACT FROM AUTHOR]

A signed graph (G, σ) is called a parity signed graph if there exists a bijective mapping f : V (G) → {1, . . ., |V (G)|} such that for each edge uv in G, f(u) and f(v) have same parity if σ(uv) = +1, and opposite parity if σ(uv) = −1. The rna number σ−(G) of G is the least number of negative edges among all possible parity signed graphs over G. Equivalently, σ−(G) is the least size of an edge-cut of G that has nearly equal sides. In this paper, we show that for the generalized Petersen graph Pn,k, σ−(Pn,k) lies between 3 and n. Moreover, we determine the exact value of σ−(Pn,k) for k ∈ {1, 2}. The rna numbers of some famous generalized Petersen graphs, namely, Petersen graph, Dürer graph, Möbius-Kantor graph, Dodecahedron, Desargues graph and Nauru graph are also computed. Recently, Acharya, Kureethara and Zaslavsky characterized the structure of those graphs whose rna number is 1. We use this characterization to show that the smallest order of a (4n + 1)-regular graph having rna number 1 is 8n + 6. We also prove the smallest order of (4n − 1)-regular graphs having rna number 1 is bounded above by 12n − 2. In particular, we show that the smallest order of a cubic graph having rna number 1 is 10. [ABSTRACT FROM AUTHOR]

A dominating set S in a graph G is called a radius dominating set if for each vertex V ∈ v -- S there is at least one vertex in u ∈ S S such that d(u, v) = rad(G). The cardinality of the minimum radius dominating set is called the radius domination number of G, denoted by γrd(G). In this article, we study the properties of this parameter. The radius domination number of some families of standard graphs are obtained and the bounds are estimated. [ABSTRACT FROM AUTHOR]

The arithmetic-geometric index AG(G) and the geometric-arithmetic index GA(G) of a graph G are defined ... and ..., where E(G) is the edge set of G, and dG(u) and dG(v) are the degrees of vertices u and v, respectively. We study relations between AG(G) and GA(G) for graphs G of given size, minimum degree and maximum degree. We present lower and upper bounds on AG(G) + GA(G), AG(G) - GA(G) and AG(G) · GA(G). All the bounds are sharp. [ABSTRACT FROM AUTHOR]

From geometric point of view, we introduced the Sombor-Wiener index of a graph and studied the basic properties of the new index. It was shown that the Sombor-Wiener index was useful in predicting the acentric factor of octane isomers. In addition, we proposed a degree-weighted Wiener index to generalize the Schultz index, the Gutman index, and the Sombor-Wiener index. Meanwhile, we gave the calculation formula of degree-weighted Wiener index for generalized Bethe trees. [ABSTRACT FROM AUTHOR]

This paper mainly studies the volume comparison in Finsler geometry under the condition that the weighted Ricci curvature Ric1 has a lower bound. By using the Laplacian comparison theorems of distance function, we characterize the growth ratio of the volume coefficients. Further, some volume comparison theorems of Bishop-Gromov type are obtained. [ABSTRACT FROM AUTHOR]

Let G(V, E) be a graph with p vertices and q edges with vertex set V and edge set E. let 'd' denote the diameter of G and d(u, v) denote the distance between the vertices u and v in G. P.Poomalai et [9]al was introduced the concept of radio analytic mean labeling in 2019.And he also indicated one more concept radio analytic mean D-distance number 2020. here we introduce a new labeling graph called Radio Antipodal Analytic mean labeling. An Radio Antipodal Analytic mean labelling of G is a function f that assigns to each vertex a nonnegative integer such that f(u) - f(v) if d(u, v) + (f(u)2 - f(v)2 (/2 = d for any two distinct vertices u, v € V(G). The Antipodal Analytic mean number of f denoted by aamn(f) is the maximum number assigned to any vertex of G. The Radio Antipodal Analytic mean number of G, denoted by Aamn(G) is the maximum value of Aamn(f) taken over all Radio Antipodal mean labelling f of G. We prove Pn, Cycle Cn, Star K1,n, Ladder Ln, n-bistar Bn,n and fan f2n+1 are the Radio Antipodal Analytic mean graphs. [ABSTRACT FROM AUTHOR]

A total coloring of a graph G is an assignment of colors to both the vertices and edges of G, such that no two adjacent or incident vertices and edges of G are assigned the same colors. In this paper, we have discussed the total coloring and total chromatic number of middle graph, total graph of twig graph and comb graph. [ABSTRACT FROM AUTHOR]

We discuss the edge chromatic number of the triangular snake graph Tn, double triangular snake graph DTn, triple triangular snake graph TTn and alternate triangular snake graph ATn. A proper edge coloring of a graphG, is an assignment of colors to all the edges of graphG so that the adjacent edges received distinct colors. The smallest number of colors needed for such coloring is known as edge chromatic number. [ABSTRACT FROM AUTHOR]

An equitable edge coloring of a graph is a proper edge coloring for which the difference between any two color classes is at most one. The minimum cardinality of G for such coloring is called equitable edge chromatic number. In this article, we determine the theorem on equitable edge coloring for sunlet graph, wheel graph and helm graph. [ABSTRACT FROM AUTHOR]

A function f is called a Skolem Difference Odd Geometric Mean labeling for the graph G(V, E) with p vertices and q edges, if it is possible to label the vertices xV with different labeling f(x) from 1,3,5......2q+1 such that the induced map f*: E(G)→{1,3,5......2q-1} defined by... [ABSTRACT FROM AUTHOR]

A set S ⊆ V(G) in a graph G is said to be a [1,2]-complementary connected dominating set, if for every vertex 𝑣 ∈ V - S, 1 ≤ |𝑁(𝑣) ∩ S| ≤ 2 and < V - S > is connected. The minimum cardinality of a [1,2]-complementary connected dominating set is called the [1,2]-complementary connected domination number and is denoted by 𝛾[1,2]𝑐𝑐 (G). In this paper, we exhibited the results based on [1,2]-complementary connected domination number for total graph. [ABSTRACT FROM AUTHOR]

The relationship between Graph Theory and Topological Space has recently developed greatly, as researchers have been able to find solutions to some problems in daily life by transforming the problem into a graph and then generating a topological space and thus facilitating reading the problem and solving it. The researchers also studied the generation of graph from topological spaces.In this article we will present two types of relations on the edges set that generate topological spaces, and we will discuss some properties of this topology, and we will study discuss the method of returning from the topological space to the graphs through using previous relationships. [ABSTRACT FROM AUTHOR]

We introduce a novel algorithm for isotropic surface remeshing which progressively eliminates obtuse triangles and improves small angles. The main novelty of the proposed approach is a simple vertex insertion scheme that facilitates the removal of large angles, and a vertex removal operation that improves the distribution of small angles. In combination with other standard local mesh operators, e.g., connectivity optimization and local tangential smoothing, our algorithm is able to remesh efficiently a low-quality mesh surface. Our approach can be applied directly or used as a post-processing step following other remeshing approaches. Our method has a similar computational efficiency to the fastest approach available, i.e., real-time adaptive remeshing [1]. In comparison with state-of-the-art approaches, our method consistently generates better results based on evaluations using different metrics. [ABSTRACT FROM AUTHOR]

This paper reports on an exploratory study in which questionnaires assessing the understanding of grade 4 mathematics concepts learnt as prescribed by Continuous Assessment Policy Statement (CAPS) were administered to 219 grade 5 learners from 20 randomly selected primary schools in the Limpopo province of South Africa at the beginning of the year 2014. This paper reports only on the understanding displayed in responses to five out of twelve questions that sought interpretation of geometric concepts in the questionnaires. This was a both quantitative and qualitative study in which structured and semi-structured interviews were conducted with six learners on the basis of their responses to gain more insight into how they constructed their geometric meanings. Results indicated that some learners did not understand the questions but would respond to questions based on a word or diagram that they associated with a particular concept. Nonetheless, 99% learners displayed understanding and familiarity with the concept of similarity. It can be recommended in this study that the interpretation in some questions be explained using African languages that were used in the learners’ previous grades. Also the use models like photos, diagrams, graphs, symbols, icons, and other visual representations can help promote the conceptual understanding of geometric concepts to learners in the intermediate phase. The study therefore concludes that for learners to interpret geometric concepts in the intermediate phase, their geometric vocabulary should be developed such that they are able to make connections between new ideas and their prior learning. [ABSTRACT FROM AUTHOR]

In this paper, we propose a simple and effective geometric model fitting method to fit and segment multi-structure data even in the presence of severe outliers. We cast the task of geometric model fitting as a representative mode-seeking problem on hypergraphs. Specifically, a hypergraph is first constructed, where the vertices represent model hypotheses and the hyperedges denote data points. The hypergraph involves higher-order similarities (instead of pairwise similarities used on a simple graph), and it can characterize complex relationships between model hypotheses and data points. In addition, we develop a hypergraph reduction technique to remove “insignificant” vertices while retaining as many “significant” vertices as possible in the hypergraph. Based on the simplified hypergraph, we then propose a novel mode-seeking algorithm to search for representative modes within reasonable time. Finally, the proposed mode-seeking algorithm detects modes according to two key elements, i.e., the weighting scores of vertices and the similarity analysis between vertices. Overall, the proposed fitting method is able to efficiently and effectively estimate the number and the parameters of model instances in the data simultaneously. Experimental results demonstrate that the proposed method achieves significant superiority over several state-of-the-art model fitting methods on both synthetic data and real images. [ABSTRACT FROM AUTHOR]

The article offers information on chapter 19 of the book "The Plaid Model," by Richard Evan Schwartz. It discusses a fix for an even rational and provides several proofs on the graph grid GA=TA(Z²). It also mentions the use of the program grid analyzer and explores the Grid Geometry Lemma and the Graph Reconstruction Lemma.

We derive a simple bijection between geometric plane perfect matchings on 2n points in convex position and triangulations on n+2 points in convex position. We then extend this bijection to monochromatic plane perfect matchings on periodically k-colored vertices and (k+2)-gonal tilings of convex point sets. These structures are related to a generalization of Temperley-Lieb algebras and our bijections provide explicit one-to-one relations between matchings and tilings. Moreover, for a given element of one class, the corresponding element of the other class can be computed in linear time. [ABSTRACT FROM AUTHOR]

The S1 model has been central in the development of the field of network geometry. It places nodes in a similarity space and connects them with a likelihood depending on an effective distance which combines similarity and popularity dimensions, with popularity directly related to the degrees of the nodes. The S1 model has been mainly studied in its homogeneous regime, in which angular coordinates are independently and uniformly scattered on the circle. We now investigate if the model can generate networks with targeted topological features and soft communities, that is, inhomogeneous angular distributions. To that end, hidden degrees must depend on angular coordinates, and we propose a method to estimate them. We conclude that the model can be topologically invariant with respect to the soft-community structure. Our results expand the scope of the model beyond the independent hidden variables limit and can have an important impact in the embedding of real-world networks. [ABSTRACT FROM AUTHOR]

We consider first-passage percolation on the configuration model. Once the network has been generated each edge is assigned an i.i.d. weight modeling the passage time of a message along this edge. Then independently two vertices are chosen uniformly at random, a sender and a recipient, and all edges along the geodesic connecting the two vertices are coloured in red (in the case that both vertices are in the same component). In this article we prove local limit theorems for the coloured graph around the recipient in the spirit of Benjamini and Schramm. We consider the explosive regime, in which case the random distances are of finite order, and the Malthusian regime, in which case the random distances are of logarithmic order. [ABSTRACT FROM AUTHOR]

Graphic pattern (e.g. geometric design) and number-based code (e.g. digital sequencing) can store and transmit complex information more efficiently than referential modes of representation. The analysis of the two genres and their relation to one another has not advanced significantly beyond a general classification based on motion-centred geometries of symmetry. This article examines an intriguing example of patchwork coverlets from the maritime societies of Oceania, where information referencing a complex genealogical system is lodged in geometric designs. By drawing attention to the interplay of graphic pattern and number-based code and its role in the knowledge economies of maritime societies, the article offers new insight into possible ways of designing a digital informational surface that captures the behaviour of an operational system, allowing both for differentiation and integration. [ABSTRACT FROM AUTHOR]

Geometric approaches to network analysis combine simply defined models with great descriptive power. In this work we provide a method for embedding directed acyclic graphs (DAG) into Minkowski spacetime using Multidimensional scaling (MDS). First we generalise the classical MDS algorithm, defined only for metrics with a Riemannian signature, to manifolds of any metric signature. We then use this general method to develop an algorithm which exploits the causal structure of a DAG to assign space and time coordinates in a Minkowski spacetime to each vertex. As in the causal set approach to quantum gravity, causal connections in the discrete graph correspond to timelike separation in the continuous spacetime. The method is demonstrated by calculating embeddings for simple models of causal sets and random DAGs, as well as real citation networks. We find that the citation networks we test yield significantly more accurate embeddings that random DAGs of the same size. Finally we suggest a number of applications in citation analysis such as paper recommendation, identifying missing citations and fitting citation models to data using this geometric approach. [ABSTRACT FROM AUTHOR]

Analytical treatment of the composition of higher-order graphs representing linear relations between variables is developed. A path formalism to deal with problems in graph theory is introduced. It is shown how paths in the composed graph representing individual contributions to variables relation can be enumerated and represented by ordinals. The method allows for one to extract partial information and gives an alternative to classical graph approach. [ABSTRACT FROM AUTHOR]

Using existing technology, we prove a Masur-Minsky style distance formula for flip-graph distance between two triangulations, expressed as a sum of the distances of the projections of these triangulations into arc graphs of the suitable subsurfaces of S. [ABSTRACT FROM AUTHOR]

We study finite ℓ-colourable structures with an underlying pregeometry. The probability measure that is used corresponds to a process of generating such structures by which colours are first randomly assigned to all 1-dimensional subspaces and then relationships are assigned in such a way that the colouring conditions are satisfied but apart from this in a random way. We can then ask what the probability is that the resulting structure, where we now forget the specific colouring of the generating process, has a given property. With this measure we get the following results: (1) A zero-one law. (2) The set of sentences with asymptotic probability 1 has an explicit axiomatisation which is presented. (3) There is a formula [ABSTRACT FROM AUTHOR]

Equistable graphs are graphs admitting positive weights on vertices such that a subset of vertices is a maximal stable set if and only if it is of total weight 1. Strongly equistable graphs are graphs such that for every [ABSTRACT FROM AUTHOR]

The crossing number cr( G) of a graph G is the minimum number of crossings in a drawing of G in the plane with no more than two edges intersecting at any point that is not a vertex. The rectilinear crossing number [ABSTRACT FROM AUTHOR]

We study the existence and the number of k-dominating independent sets in certain graph families. While the case [ABSTRACT FROM AUTHOR]

LetA(X) be an intermediate subring ofC(X) andIbe an ideal ofA(X). We give some topological characterizations for the smallestz-idealIz(A) containingIand the largestz-idealIz(A) contained inI. Using this, we generalize some facts in the context ofC(X) to the intermediate subrings ofC(X). Furthermore, by investigating some properties of subrings of the formI+C∗(X), whereIis an ideal ofC(X), we show that every intermediateC-algebra ofC(X) is generated by a family of subrings of the formI+C∗(X). Moreover, we characterize realcompactifications ofX(specially, those which have compact remainder) via the subringsI+C∗(X). [ABSTRACT FROM AUTHOR]

Let G be a graph of order n satisfying that there exists [ABSTRACT FROM AUTHOR]

We prove that it is consistent that $\aleph _\omega $ is strong limit, $2^{\aleph _\omega } $ is large and the universality number for graphs on $\aleph _{\omega + 1} $ is small. The proof uses Prikry forcing with interleaved collapsing. [ABSTRACT FROM PUBLISHER]

It is shown that Halin graphs are $$\varDelta $$ -edge-choosable and that graphs of tree-width 3 are $$(\varDelta +1)$$ -edge-choosable and $$(\varDelta +2)$$ -total-colourable. [ABSTRACT FROM AUTHOR]

The packing and covering problems have been considered for several classes of graphs. For instance, Bryant et. al. have investigated the packing problem for paths and cycles, and the packing and covering problems for 3-cubes. The packing and covering problems were settled for stars with up to six edges by Roditty. In this paper, for every possible leave graph (excess graph), we find a corresponding maximum packing (minimum covering) of the complete graph with stars with up to five edges. [ABSTRACT FROM AUTHOR]

A graph G is free ( a, b)-choosable if for any vertex v with b colors assigned and for any list of colors of size a associated with each vertex $$u\ne v$$ , the coloring can be completed by choosing for u a subset of b colors such that adjacent vertices are colored with disjoint color sets. In this note, a necessary and sufficient condition for a cycle to be free ( a, b)-choosable is given. As a corollary, we obtain almost optimal results about the free ( a, b)-choosability of outerplanar graphs. [ABSTRACT FROM AUTHOR]

Let G be a plane graph. A vertex-colouring $$\varphi $$ of G is called facial non-repetitive if for no sequence $$r_1 r_2 \ldots r_{2n}$$ , $$n\ge 1$$ , of consecutive vertex colours of any facial path it holds $$r_i=r_{n+i}$$ for all $$i=1,2,\ldots ,n$$ . A plane graph G is facial non-repetitively k -choosable if for every list assignment $$L:V\rightarrow 2^{\mathbb {N}}$$ with minimum list size at least k there is a facial non-repetitive vertex-colouring $$\varphi $$ with colours from the associated lists. The facial Thue choice number, $$\pi _{fl}(G)$$ , of a plane graph G is the minimum number k such that G is facial non-repetitively k-choosable. We use the so-called entropy compression method to show that $$\pi _{fl} (G)\le c \varDelta $$ for some absolute constant c and G a plane graph with maximum degree $$\varDelta $$ . Moreover, we give some better (constant) upper bounds on $$\pi _{fl} (G)$$ for special classes of plane graphs. [ABSTRACT FROM AUTHOR]

Reciprocal screw theory is used to recognize the kinematic joints of assemblies restricted by arbitrary combinations of geometry constraints. Kinematic analysis is common for reaching a satisfactory design. If a machine is large and the incidence of redesign frequent is high, then it becomes imperative to have fast analysis-redesign-reanalysis cycles. This work addresses this problem by providing recognition technology for converting a 3D assembly model into a kinematic joint model, which is represented by a graph of parts with kinematic joints among them. The three basic components of the geometric constraints are described in terms of wrench, and it is thus easy to model each common assembly constraint. At the same time, several different types of kinematic joints in practice are presented in terms of twist. For the reciprocal product of a twist and wrench, which is equal to zero, the geometry constraints can be converted into the corresponding kinematic joints as a result. To eliminate completely the redundant components of different geometry constraints that act upon the same part, the specific operation of a matrix space is applied. This ability is useful in supporting the kinematic design of properly constrained assemblies in CAD systems. [ABSTRACT FROM AUTHOR]

Let P be a finite set of points in general position in the plane. The structure of the complete graph K( P) as a geometric graph includes, for any pair [ a, b], [ c, d] of vertex-disjoint edges, the information whether they cross or not. The simple (i.e., non-crossing) spanning trees (SSTs) of K( P) are the vertices of the so-called Geometric Tree Graph of P, G( P). Two such vertices are adjacent in G( P) if they differ in exactly two edges, i.e., if one can be obtained from the other by deleting an edge and adding another edge. In this paper we show how to reconstruct from G( P) (regarded as an abstract graph) the structure of K( P) as a geometric graph. We first identify within G( P) the vertices that correspond to spanning stars. Then we regard each star S( z) with center z as the representative in G( P) of the vertex z of K( P). (This correspondence is determined only up to an automorphism of K( P) as a geometric graph.) Finally we determine for any four distinct stars S( a), S( b), S( c), and S( d), by looking at their relative positions in G( P), whether the corresponding segments cross. [ABSTRACT FROM AUTHOR]