P3F92-: An All-Pseudo-pi* 2 pi-Aromatic

A qualitative MO analysis suggests (PH3)(3)(2-) as a candidate for an all-pseudo-pi* 2 pi-aromatic; however computational studies rule out its existence. Fluorine substitution which increases the contribution of p orbitals on P in the pseudo-pi* MO makes (PF3)(3)(2-) a minimum and an aromatic. The 2 pi aromaticity arising from the bonding combination of the three pseudo-pi* fragment MOs is comparable to that in C3O32- and is another example for the analogy between CO and PF3. The dianion (PF3)(3)(2-) forms the first example of a three-membered ring with all the vertices constituted by pentacoordinate phosphorus. The ability of PF3 to form the all-pseudo-pi* 2 pi-aromatic system is not shared by the heavier analogues, AsF3 and SbF3.

Boosting Higgs CP properties via VH production at the Large Hadron Collider

We consider ZH and WH production at the Large Hadron Collider, where the Higgs decays to a b (b) over bar pair. We use jet substructure techniques to reconstruct the Higgs boson and construct angular observables involving leptonic decay products of the vector bosons. These efficiently discriminate between the tensor structure of the HVV vertex expected in the Standard Model and that arising from possible new physics, as quantified by higher dimensional operators. This can then be used to examine the CP nature of the Higgs as well as CP mixing effects in the HZZ and HWW vertices separately. (C) 2014 Elsevier B.V.

Freeform Skeletal Shape Optimization of Compliant Mechanisms

Compliant mechanisms are elastic continua used to transmit or transform force and motion mechanically. The topology optimization methods developed for compliant mechanisms also give the shape for a chosen parameterization of the design domain with a fixed mesh. However, in these methods, the shapes of the flexible segments in the resulting optimal solutions are restricted either by the type or the resolution of the design parameterization. This limitation is overcome in this paper by focusing on optimizing the skeletal shape of the compliant segments in a given topology. It is accomplished by identifying such segments in the topology and representing them using Bezier curves. The vertices of the Bezier control polygon are used to parameterize the shape-design space. Uniform parameter steps of the Bezier curves naturally enable adaptive finite element discretization of the segments as their shapes change. Practical constraints such as avoiding intersections with other segments, self-intersections, and restrictions on the available space and material, are incorporated into the formulation. A multi-criteria function from our prior work is used as the objective. Analytical sensitivity analysis for the objective and constraints is presented and is used in the numerical optimization. Examples are included to illustrate the shape optimization method.

Subexponential Algorithm for d-Cluster Edge Deletion: Exception or Rule?

The correlation clustering problem is a fundamental problem in both theory and practice, and it involves identifying clusters of objects in a data set based on their similarity. A traditional modeling of this question as a graph theoretic problem involves associating vertices with data points and indicating similarity by adjacency. Clusters then correspond to cliques in the graph. The resulting optimization problem, Cluster Editing (and several variants) are very well-studied algorithmically. In many situations, however, translating clusters to cliques can be somewhat restrictive. A more flexible notion would be that of a structure where the vertices are mutually ``not too far apart'', without necessarily being adjacent. One such generalization is realized by structures called s-clubs, which are graphs of diameter at most s. In this work, we study the question of finding a set of at most k edges whose removal leaves us with a graph whose components are s-clubs. Recently, it has been shown that unless Exponential Time Hypothesis fail (ETH) fails Cluster Editing (whose components are 1-clubs) does not admit sub-exponential time algorithm STACS, 2013]. That is, there is no algorithm solving the problem in time 2 degrees((k))n(O(1)). However, surprisingly they show that when the number of cliques in the output graph is restricted to d, then the problem can be solved in time O(2(O(root dk)) + m + n). We show that this sub-exponential time algorithm for the fixed number of cliques is rather an exception than a rule. Our first result shows that assuming the ETH, there is no algorithm solving the s-Club Cluster Edge Deletion problem in time 2 degrees((k))n(O(1)). We show, further, that even the problem of deleting edges to obtain a graph with d s-clubs cannot be solved in time 2 degrees((k))n(O)(1) for any fixed s, d >= 2. This is a radical contrast from the situation established for cliques, where sub-exponential algorithms are known.

REPRESENTING A CUBIC GRAPH AS THE INTERSECTION GRAPH OF AXIS-PARALLEL BOXES IN THREE DIMENSIONS

We show that every graph of maximum degree 3 can be represented as the intersection graph of axis parallel boxes in three dimensions, that is, every vertex can be mapped to an axis parallel box such that two boxes intersect if and only if their corresponding vertices are adjacent. In fact, we construct a representation in which any two intersecting boxes touch just at their boundaries.

BELIEF PROPAGATION FOR OPTIMAL EDGE COVER IN THE RANDOM COMPLETE GRAPH

We apply the objective method of Aldous to the problem of finding the minimum-cost edge cover of the complete graph with random independent and identically distributed edge costs. The limit, as the number of vertices goes to infinity, of the expected minimum cost for this problem is known via a combinatorial approach of Hessler and Wastlund. We provide a proof of this result using the machinery of the objective method and local weak convergence, which was used to prove the (2) limit of the random assignment problem. A proof via the objective method is useful because it provides us with more information on the nature of the edge's incident on a typical root in the minimum-cost edge cover. We further show that a belief propagation algorithm converges asymptotically to the optimal solution. This can be applied in a computational linguistics problem of semantic projection. The belief propagation algorithm yields a near optimal solution with lesser complexity than the known best algorithms designed for optimality in worst-case settings.

A constant factor approximation algorithm for boxicity of circular arc graphs

The boxicity (resp. cubicity) of a graph G(V, E) is the minimum integer k such that G can be represented as the intersection graph of axis parallel boxes (resp. cubes) in R-k. Equivalently, it is the minimum number of interval graphs (resp. unit interval graphs) on the vertex set V, such that the intersection of their edge sets is E. The problem of computing boxicity (resp. cubicity) is known to be inapproximable, even for restricted graph classes like bipartite, co-bipartite and split graphs, within an O(n(1-epsilon))-factor for any epsilon > 0 in polynomial time, unless NP = ZPP. For any well known graph class of unbounded boxicity, there is no known approximation algorithm that gives n(1-epsilon)-factor approximation algorithm for computing boxicity in polynomial time, for any epsilon > 0. In this paper, we consider the problem of approximating the boxicity (cubicity) of circular arc graphs intersection graphs of arcs of a circle. Circular arc graphs are known to have unbounded boxicity, which could be as large as Omega(n). We give a (2 + 1/k) -factor (resp. (2 + log n]/k)-factor) polynomial time approximation algorithm for computing the boxicity (resp. cubicity) of any circular arc graph, where k >= 1 is the value of the optimum solution. For normal circular arc (NCA) graphs, with an NCA model given, this can be improved to an additive two approximation algorithm. The time complexity of the algorithms to approximately compute the boxicity (resp. cubicity) is O(mn + n(2)) in both these cases, and in O(mn + kn(2)) = O(n(3)) time we also get their corresponding box (resp. cube) representations, where n is the number of vertices of the graph and m is its number of edges. Our additive two approximation algorithm directly works for any proper circular arc graph, since their NCA models can be computed in polynomial time. (C) 2014 Elsevier B.V. All rights reserved.

Fixed-orientation equilateral triangle matching of point sets

Given a point set P and a class C of geometric objects, G(C)(P) is a geometric graph with vertex set P such that any two vertices p and q are adjacent if and only if there is some C is an element of C containing both p and q but no other points from P. We study G(del)(P) graphs where del is the class of downward equilateral triangles (i.e., equilateral triangles with one of their sides parallel to the x-axis and the corner opposite to this side below that side). For point sets in general position, these graphs have been shown to be equivalent to half-Theta(6) graphs and TD-Delaunay graphs. The main result in our paper is that for point sets P in general position, G(del)(P) always contains a matching of size at least vertical bar P vertical bar-1/3] and this bound is tight. We also give some structural properties of G(star)(P) graphs, where is the class which contains both upward and downward equilateral triangles. We show that for point sets in general position, the block cut point graph of G(star)(P) is simply a path. Through the equivalence of G(star)(P) graphs with Theta(6) graphs, we also derive that any Theta(6) graph can have at most 5n-11 edges, for point sets in general position. (C) 2013 Elsevier B.V. All rights reserved.

2-Connecting outerplanar graphs without blowing up the pathwidth

Given a connected outerplanar graph G of pathwidth p, we give an algorithm to add edges to G to get a supergraph of G, which is 2-vertex-connected, outerplanar and of pathwidth O(p). This settles an open problem raised by Biedl 1], in the context of computing minimum height planar straight line drawings of outerplanar graphs, with their vertices placed on a two-dimensional grid. In conjunction with the result of this paper, the constant factor approximation algorithm for this problem obtained by Biedl 1] for 2-vertex-connected outerplanar graphs will work for all outer planar graphs. (C) 2014 Elsevier B.V. All rights reserved.

Parameterized Algorithms for MAX COLORABLE INDUCED SUBGRAPH Problem on Perfect Graphs

We address the parameterized complexity ofMaxColorable Induced Subgraph on perfect graphs. The problem asks for a maximum sized q-colorable induced subgraph of an input graph G. Yannakakis and Gavril IPL 1987] showed that this problem is NP-complete even on split graphs if q is part of input, but gave a n(O(q)) algorithm on chordal graphs. We first observe that the problem is W2]-hard parameterized by q, even on split graphs. However, when parameterized by l, the number of vertices in the solution, we give two fixed-parameter tractable algorithms. The first algorithm runs in time 5.44(l) (n+#alpha(G))(O(1)) where #alpha(G) is the number of maximal independent sets of the input graph. The second algorithm runs in time q(l+o()l())n(O(1))T(alpha) where T-alpha is the time required to find a maximum independent set in any induced subgraph of G. The first algorithm is efficient when the input graph contains only polynomially many maximal independent sets; for example split graphs and co-chordal graphs. The running time of the second algorithm is FPT in l alone (whenever T-alpha is a polynomial in n), since q <= l for all non-trivial situations. Finally, we show that (under standard complexitytheoretic assumptions) the problem does not admit a polynomial kernel on split and perfect graphs in the following sense: (a) On split graphs, we do not expect a polynomial kernel if q is a part of the input. (b) On perfect graphs, we do not expect a polynomial kernel even for fixed values of q >= 2.

Rainbow Connection Number of Graph Power and Graph Products

Rainbow connection number, rc(G), of a connected graph G is the minimum number of colors needed to color its edges so that every pair of vertices is connected by at least one path in which no two edges are colored the same (note that the coloring need not be proper). In this paper we study the rainbow connection number with respect to three important graph product operations (namely the Cartesian product, the lexicographic product and the strong product) and the operation of taking the power of a graph. In this direction, we show that if G is a graph obtained by applying any of the operations mentioned above on non-trivial graphs, then rc(G) a parts per thousand currency sign 2r(G) + c, where r(G) denotes the radius of G and . In general the rainbow connection number of a bridgeless graph can be as high as the square of its radius 1]. This is an attempt to identify some graph classes which have rainbow connection number very close to the obvious lower bound of diameter (and thus the radius). The bounds reported are tight up to additive constants. The proofs are constructive and hence yield polynomial time -factor approximation algorithms.

LOWER BOUNDS FOR BOXICITY

An axis-parallel b-dimensional box is a Cartesian product R-1 x R-2 x ... x R-b where R-i is a closed interval of the form a(i),b(i)] on the real line. For a graph G, its boxicity box(G) is the minimum dimension b, such that G is representable as the intersection graph of boxes in b-dimensional space. Although boxicity was introduced in 1969 and studied extensively, there are no significant results on lower bounds for boxicity. In this paper, we develop two general methods for deriving lower bounds. Applying these methods we give several results, some of which are listed below: 1. The boxicity of a graph on n vertices with no universal vertices and minimum degree delta is at least n/2(n-delta-1). 2. Consider the g(n,p) model of random graphs. Let p <= 1 - 40logn/n(2.) Then with high `` probability, box(G) = Omega(np(1 - p)). On setting p = 1/2 we immediately infer that almost all graphs have boxicity Omega(n). Another consequence of this result is as follows: For any positive constant c < 1, almost all graphs on n vertices and m <= c((n)(2)) edges have boxicity Omega(m/n). 3. Let G be a connected k-regular graph on n vertices. Let lambda be the second largest eigenvalue in absolute value of the adjacency matrix of G. Then, the boxicity of G is a least (kappa(2)/lambda(2)/log(1+kappa(2)/lambda(2))) (n-kappa-1/2n). 4. For any positive constant c 1, almost all balanced bipartite graphs on 2n vertices and m <= cn(2) edges have boxicity Omega(m/n).

Rainbow matchings in strongly edge-colored graphs

A rainbow matching of an edge-colored graph G is a matching in which no two edges have the same color. There have been several studies regarding the maximum size of a rainbow matching in a properly edge-colored graph G in terms of its minimum degree 3(G). Wang (2011) asked whether there exists a function f such that a properly edge-colored graph G with at least f (delta(G)) vertices is guaranteed to contain a rainbow matching of size delta(G). This was answered in the affirmative later: the best currently known function Lo and Tan (2014) is f(k) = 4k - 4, for k >= 4 and f (k) = 4k - 3, for k <= 3. Afterwards, the research was focused on finding lower bounds for the size of maximum rainbow matchings in properly edge-colored graphs with fewer than 4 delta(G) - 4 vertices. Strong edge-coloring of a graph G is a restriction of proper edge-coloring where every color class is required to be an induced matching, instead of just being a matching. In this paper, we give lower bounds for the size of a maximum rainbow matching in a strongly edge-colored graph Gin terms of delta(G). We show that for a strongly edge-colored graph G, if |V(G)| >= 2 |3 delta(G)/4|, then G has a rainbow matching of size |3 delta(G)/4|, and if |V(G)| < 2 |3 delta(G)/4|, then G has a rainbow matching of size |V(G)|/2] In addition, we prove that if G is a strongly edge-colored graph that is triangle-free, then it contains a rainbow matching of size at least delta(G). (C) 2015 Elsevier B.V. All rights reserved.

Strongly minimal triangulations of (S (3) x S (1))(#3) and (S (3) x S (1))(#3)

A triangulated d-manifold K, satisfies the inequality for da parts per thousand yen3. The triangulated d-manifolds that meet the bound with equality are called tight neighbourly. In this paper, we present tight neighbourly triangulations of 4-manifolds on 15 vertices with as an automorphism group. One such example was constructed by Bagchi and Datta (Discrete Math. 311 (citeyearbd102011) 986-995). We show that there are exactly 12 such triangulations up to isomorphism, 10 of which are orientable.

SEPARATION DIMENSION OF BOUNDED DEGREE GRAPHS

The separation dimension of a graph G is the smallest natural number k for which the vertices of G can be embedded in R-k such that any pair of disjoint edges in G can be separated by a hyperplane normal to one of the axes. Equivalently, it is the smallest possible cardinality of a family F of total orders of the vertices of G such that for any two disjoint edges of G, there exists at least one total order in F in which all the vertices in one edge precede those in the other. In general, the maximum separation dimension of a graph on n vertices is Theta(log n). In this article, we focus on bounded degree graphs and show that the separation dimension of a graph with maximum degree d is at most 2(9) (log*d)d. We also demonstrate that the above bound is nearly tight by showing that, for every d, almost all d-regular graphs have separation dimension at least d/2]