Small Number Of Vertices Research Articles

Asymmetry of Convex Polytopes and Vertex Index of Symmetric Convex Bodies

In (Gluskin, Litvak in Geom. Dedicate 90:45–48, [2002]) it was shown that a polytope with few vertices is far from being symmetric in the Banach–Mazur distance. More precisely, it was shown that Banach–Mazur distance between such a polytope and any symmetric convex body is large. In this note we introduce a new, averaging-type parameter to measure the asymmetry of polytopes. It turns out that, surprisingly, this new parameter is still very large, in fact it satisfies the same lower bound as the Banach–Mazur distance. In a sense it shows the following phenomenon: if a convex polytope with small number of vertices is as close to a symmetric body as it can be, then most of its vertices are as bad as the worst one. We apply our results to provide a lower estimate on the vertex index of a symmetric convex body, which was recently introduced in (Bezdek, Litvak in Adv. Math. 215:626–641, [2007]). Furthermore, we give the affirmative answer to a conjecture by Bezdek (Period. Math. Hung. 53:59–69, [2006]) on the quantitative illumination problem.

On geometric constructions of (k, g)-graphs

We give new constructions for k-regular graphs of girth 6, 8 and 12 with a small number of vertices. The key idea is to start with a generalized n-gon and delete some lines and points to decrease the valency of the incidence graph.

Nonflexible polyhedra with a small number of vertices

The paper shows which embedded and immersed polyhedra with no more than eight vertices are nonflexible. It turns out that all embedded polyhedra are nonflexible, except possibly for polyhedra of one of the combinatorial types, for which the problem still remains open.

Incidence Matrices of Projective Planes and of Some Regular Bipartite Graphs of Girth 6 with Few Vertices

Let $q$ be a prime power and $r=0,1\ldots, q-3$. Using the Latin squares obtained by multiplying each entry of the addition table of the Galois field of order $q$ by an element distinct from zero, we obtain the incidence matrices of projective planes and the incidence matrices of $(q-r)$-regular bipartite graphs of girth 6 and $q^2-rq-1$ vertices in each partite set. Moreover, in this work two Latin squares of order $q-1$ with entries belonging to $\{0,1,\ldots, q\}$, not necessarily the same, are defined to be quasi row-disjoint if and only if the Cartesian product of any two rows contains at most one pair $(x,x)$ with $x\ne 0$. Using these quasi row-disjoint Latin squares we find $(q-1)$-regular bipartite graphs of girth 6 with $q^2-q-2$ vertices in each partite set. Some of these graphs have the smallest number of vertices known so far among the regular graphs with girth 6.

Homomorphisms of 2-edge-colored graphs

In this paper, we study homomorphisms of 2-edge-colored graphs, that is graphs with edges colored with two colors. We consider various graph classes (outerplanar graphs, partial 2-trees, partial 3-trees, planar graphs) and the problem is to find, for each class, the smallest number of vertices of a 2-edge-colored graph H such that each graph of the considered class admits a homomorphism to H.

On the vertex index of convex bodies

We introduce the vertex index, vein ( K ) , of a given centrally symmetric convex body K ⊂ R d , which, in a sense, measures how well K can be inscribed into a convex polytope with small number of vertices. This index is closely connected to the illumination parameter of a body, introduced earlier by the first named author, and, thus, related to the famous conjecture in Convex Geometry about covering of a d-dimensional body by 2 d smaller positively homothetic copies. We provide asymptotically sharp estimates (up to a logarithmic term) of this index in the general case. More precisely, we show that for every centrally symmetric convex body K ⊂ R d one has d 3 / 2 2 π e ovr ( K ) ⩽ vein ( K ) ⩽ C d 3 / 2 ln ( 2 d ) , where ovr ( K ) = inf ( vol ( E ) / vol ( K ) ) 1 / d is the outer volume ratio of K with the infimum taken over all ellipsoids E ⊃ K and with vol ( ⋅ ) denoting the volume. Also, we provide sharp estimates in dimensions 2 and 3. Namely, in the planar case we prove that 4 ⩽ vein ( K ) ⩽ 6 with equalities for parallelograms and affine regular convex hexagons, and in the 3-dimensional case we show that 6 ⩽ vein ( K ) with equality for octahedra. We conjecture that the vertex index of a d-dimensional Euclidean ball (respectively ellipsoid) is 2 d d . We prove this conjecture in dimensions two and three.

On the Computational Complexity of the Forcing Chromatic Number

We consider vertex colorings of graphs in which adjacent vertices have distinct colors. A graph is $s$-chromatic if it is colorable in $s$ colors and any coloring of it uses at least $s$ colors. The forcing chromatic number $F_{\chi}(G)$ of an $s$-chromatic graph $G$ is the smallest number of vertices which must be colored so that, with the restriction that $s$ colors are used, every remaining vertex has its color determined uniquely. We estimate the computational complexity of $\force G$ relating it to the complexity class US introduced by Blass and Gurevich [Inform. Control, 55 (1982), pp. 80-88]. We prove that recognizing whether $F_{\chi}(G)\le2$ is US-hard with respect to polynomial-time many-one reductions. Moreover, this problem is coNP-hard even under the promises that $F_{\chi}(G)\le3$ and $G$ is 3-chromatic. On the other hand, recognizing whether $F_{\chi}(G)\le k$, for each constant $k$, is reducible to a problem in US via a disjunctive truth-table reduction. Similar results are obtained also for forcing variants of the clique and the domination numbers of a graph.

Regular heptagon's intersections circles

The regular heptagon (i.e., the planar regular convex polygon with seven vertices) has not been studied extensively like its cousins the equilateral triangle, the square, the regular pentagon, and the regular hexagon. Perhaps the reason is because this is the regular polygon with the smallest number of vertices that cannot be constructed only with compass and straightedge. The few sporadic known results on regular heptagons were reviewed by Leon Bankoff and Jack Garfunkel 30 years ago in the reference [1].

Using expander graphs to find vertex connectivity

The (vertex) connectivity κ of a graph is the smallest number of vertices whose deletion separates the graph or makes it trivial. We present the fastest known algorithm for finding κ. For a digraph with n vertices, m edges and connectivity κ the time bound is O (( n + min{κ 5/2; , κ n 3/4; }) m ). This improves the previous best bound of O (( n + min{κ 3 , κ n }) m ). For an undirected graph both of these bounds hold with m replaced by κ n . Expander graphs are useful for solving the following subproblem that arises in connectivity computation: A known set R of vertices contains two large but unknown subsets that are separated by some unknown set S of κ vertices; we must find two vertices of R that are separated by S .

POSE ESTIMATION IN AUTOMATED VISUAL INSPECTION USING GENETIC ALGORITHM

In this paper, we propose a genetic algorithm based approach to determine the pose of an object in Automated Visual Inspection having three degrees of freedom. We have investigated the effect of noise at 20 dB SNR and also mismatch resulting from incorrect correspondences between the object space points and the image space points, on the estimation of pose parameters. The maximum error in translation parameters is less than 0.45 cm and rotational error is less than 0.2 degree at 20 dB SNR. The error in parameter estimation is insignificant upto 7 pairs of mismatched points out of 24 points in object space and the results skyrockets when 8 or more pairs of points are mismatched. We have compared our result with that obtained by least square technique and it shows that GA based method outperform the gradient based technique when the number of vertices of the object to be inspected is small. These results have clearly established the robustness of GA in estimating the pose of an object with small number of vertices in automated visual inspection.

Cut loci in lens manifolds

Cut loci in geometric three-manifolds equipped with their natural metrics are an interesting source of spines with small number of vertices. An application of this principle to lens manifolds reveals an interplay between their geometry and topology, combinatorial types of convex hulls of group orbits, and estimates of rotation distance between certain triangulations. To cite this article: S. Anisov, C. R. Acad. Sci. Paris, Ser. I 342 (2006).

Decycling Cartesian Products of Two Cycles

The decycling number $\nabla(G)$ of a graph G is the smallest number of vertices which can be removed from G so that the resultant graph contains no cycles. In this paper, we study the decycling number for the family of graphs consisting of the Cartesian product of two cycles. We completely solve the problem of determining the decycling number of $C_m \square C_n$ for all m and n. Moreover, we find a vertex set T that yields a maximum induced tree in $C_m\square C_n$.

Networking genetic regulation and neural computation: Directed network topology and its effect on the dynamics

Two different types of directed networks are investigated, transcriptional regulation networks and neural networks. The directed network structure is studied and is also shown to reflect the different processes taking place on the networks. The distribution of influence, identified as the the number of downstream vertices, are used as a tool for investigating random vertex removal. In the transcriptional regulation networks we observe that only a small number of vertices have a large influence. The small influences of most vertices limit the effect of a random removal to, in most cases, only a small fraction of vertices in the network. The neural network has a rather different topology with respect to the influence, which are large for most vertices. To further investigate the effect of vertex removal we simulate the biological processes taking place on the networks. Opposed to the presumed large effect of random vertex removal in the neural network, the high density of edges in conjunction with the dynamics used makes the change in the state of the system to be highly localized around the removed vertex.

Decycling numbers of random regular graphs

AbstractThe decycling number ϕ(G) of a graph G is the smallest number of vertices which can be removed from G so that the resultant graph contains no cycles. In this paper, we study the decycling numbers of random regular graphs. For a random cubic graph G of order n, we prove that ϕ(G) = ⌈n/4 + 1/2⌉ holds asymptotically almost surely. This is the result of executing a greedy algorithm for decycling G making use of a randomly chosen Hamilton cycle. For a general random d‐regular graph G of order n, where d ≥ 4, we prove that ϕ(G)/n can be bounded below and above asymptotically almost surely by certain constants b(d) and B(d), depending solely on d, which are determined by solving, respectively, an algebraic equation and a system of differential equations. © 2002 Wiley Periodicals, Inc. Random Struct. Alg., 21: 397–413, 2002

Graphs having distance- n domination number half their order

For any positive integer n and any graph G a set D of vertices of G is a distance- n dominating set, if every vertex v∈ V( G)− D has exactly distance n to at least one vertex in D. The distance- n domination number γ = n ( G) is the smallest number of vertices in any distance- n dominating set. If G is a graph of order p and each vertex in G has distance n to at least one vertex in G, then the distance- n domination number has the upper bound p/2 as Ore's upper bound on the classical domination number. In this paper, a characterization is given for graphs having distance- n domination number equal to half their order, when the diameter is greater or equal 2 n−1. With this result we confirm a conjecture of Boland, Haynes, and Lawson.

Block graphs with unique minimum dominating sets

For any graph G a set D of vertices of G is a dominating set, if every vertex v∈ V( G)− D has at least one neighbor in D. The domination number γ( G) is the smallest number of vertices in any dominating set. In this paper, a characterization is given for block graphs having a unique minimum dominating set. With this result, we generalize a theorem of Gunther, Hartnell, Markus and Rall for trees.

The packing property

A clutter (V, E) packs if the smallest number of vertices needed to intersect all the edges (i.e. a minimum transversal) is equal to the maximum number of pairwise disjoint edges (i.e. a maximum matching). This terminology is due to Seymour 1977. A clutter is minimally nonpacking if it does not pack but all its minors pack. An m×n 0,1 matrix is minimally nonpacking if it is the edge-vertex incidence matrix of a minimally nonpacking clutter. Minimally nonpacking matrices can be viewed as the counterpart for the set covering problem of minimally imperfect matrices for the set packing problem. This paper proves several properties of minimally nonpacking clutters and matrices.

Efficient Algorithms for Approximating Polygonal Chains

We consider the problem of approximating a polygonal chain C by another polygonal chain C' whose vertices are constrained to be a subset of the set of vertices of C . The goal is to minimize the number of vertices needed in the approximation C' . Based on a framework introduced by Imai and Iri [25], we define an error criterion for measuring the quality of an approximation. We consider two problems. (1) Given a polygonal chain C and a parameter ɛ \geq 0 , compute an approximation of C , among all approximations whose error is at most ɛ , that has the smallest number of vertices. We present an O(n4/3 + δ) -time algorithm to solve this problem, for any δ > 0; the constant of proportionality in the running time depends on δ . (2) Given a polygonal chain C and an integer k , compute an approximation of C with at most k vertices whose error is the smallest among all approximations with at most k vertices. We present a simple randomized algorithm, with expected running time O(n4/3 + δ) , to solve this problem.

Computing Vertex Connectivity: New Bounds from Old Techniques

The vertex connectivity κ of a graph is the smallest number of vertices whose deletion separates the graph or makes it trivial. We present the fastest known deterministic algorithm for finding the vertex connectivity and a corresponding separator. The time for a digraph having n vertices and m edges is O(min{κ3+n,κn}m); for an undirected graph the term m can be replaced by κn. A randomized algorithm finds κ with error probability 1/2 in time O(nm). If the vertices have nonnegative weights the weighted vertex connectivity is found in time O(κ1nmlog(n2/m)) where κ1≤m/n is the unweighted vertex connectivity or in expected time O(nmlog(n2/m)) with error probability 1/2. The main algorithm combines two previous vertex connectivity algorithms and a generalization of the preflow-push algorithm of Hao and Orlin (1994, J. Algorithms17, 424–446) that computes edge connectivity.

7. Testing Transitivity in Digraphs

The problem of testing the transitivity of a relationship observed in a digraph, taking as many nontransitivity related irregularities as possible into account, is studied. Two test quantities are used: (1) the proportion of transitive triples out of all nonvacuously transitive triples, and (2) the density difference (the difference between mean local transitivity density and overall edge density). The null distribution used is the rather complex uniform distribution on digraphs conditional on the indegrees and outdegrees. A simulation study is made in order to estimate critical values of the tests for different significance levels. When all vertices have the same indegree and outdegree, the occurrence of transitive triples is rather infrequent in most conditional uniform graphs; this is reflected by low critical values of the transitivity-related test statistics. When both the indegree and outdegree sequences are skewed in the same direction, there is a small number of vertices with large indegrees and outdegrees. This results in a clustering structure, in which transitive triples occur frequently; in such conditional uniform graphs, the critical values of the test statistics are rather high. The powers of the tests are estimated against the Bernoulli transitive triple model, which assumes a simple random graph distribution in which the transitivity is high. The test based on density difference has the highest power in many cases. The tests are applied to a large set of classroom sociograms, and in this situation it is also found that uniform randomness is rejected in favor of transitivity most frequently when the test based on the density difference is used. However, the vast majority of these sociograms are so far from the uniform distribution that the null hypothesis of uniform randomness is rejected regardless of which test is used. Nevertheless, the results imply that the density difference is the best detector of transitivity related to the measures examined.