EXTREMAL PROBLEMS CONCERNING CYCLES IN GRAPHS AND THEIR COMPLEMENTS

Let Gt(n) be the class of connected graphs on n vertices having the longest cycle of length t and let G ∈ Gt(n). Woodall (1976) determined the maximum number of edges of G, ε(G) ≤ w(n,t), where w(n, t) = (n - 1) t/2 - r(t – r - 1)/2 and r = (n - 1 ) - (t - 1) ⎣(n - 1)/(t - 1)⎦. An alternative proof and characterization of the extremal (edge-maximal) graphs given by Caccetta and Vijayan (1991). The edge- maximal graphs have the property that their complements are either disconnected or have a cycle going through each vertex (i.e. they are hamiltonian). This motivates us to investigate connected graphs with prescribed circumference (length of the longest cycle) having connected complements with cycles . More specifically, we focus our investigations on : Let G(n, c, c ) denote the class of connected graphs on n vertices having circumference c and whose connected complements have circumference c . The problem of interest is that of determining the bounds of the number of edges of a graph G ∈ G(n, c, c ) and characterize the extremal graphs of G(n, c, c ). We discuss the class G(n, c, c ) and present some results for small c. In particular for c = 4 and c = n - 2, we provide a complete solution. Key words : extremal graph, circumferenc

The Modified CW1 Algorithm for the Degree Restricted Minimum Spanning Tree Problem

Given edge weighted graph G (all weights are non-negative), The Degree Constrained Minimum Spanning Tree Problem is concerned with finding the minimum weight spanning tree T satisfying specified degree restrictions on the vertices. This problem arises naturally in communication networks where the degree of a vertex represents the number of line interfaces available at a terminal (center). The applications of the Degree Constrained Minimum Spanning Tree problems that may arise in real-life include: the design of telecommunication, transportation, and energy networks. It is also used as a subproblem in the design of networks for computer communication, transportation, sewage and plumbing. Since, apart from some trivial cases, the problem is computationally difficult (NP-complete), a number of heuristics have been proposed. In this paper we will discuss the modification of CW1 Algorithm that already proposed by Wamiliana and Caccetta (2003). The results on540 random table problems will be discussed

Accuracy assessment of a 300 m global land cover map : the GlobCover experience

The GlobCover project supported by ESA has developed an operational service dedicated to the generation of global land cover maps through an automated classification of MERIS FRS time series. This paper reports the independent accuracy assessment of the global GlobCover product as the first global exercise implemented according to the CEOS Land Product Validation group recommendations. Based on a network of 16 international experts and on-line tools, a unique, globally distributed reference data set was collected in a standardized manner and used to derive mapping accuracy figures. The overall accuracy, weighted by the area proportions of the various land cover classes, is 73 % based on a set of 3167 samples. These results are discussed with regards to the previous experiences. (Résumé d'auteur

Polylogarithmic Supports are required for Approximate Well-Supported Nash Equilibria below 2/3

In an epsilon-approximate Nash equilibrium, a player can gain at most epsilon in expectation by unilateral deviation. An epsilon well-supported approximate Nash equilibrium has the stronger requirement that every pure strategy used with positive probability must have payoff within epsilon of the best response payoff. Daskalakis, Mehta and Papadimitriou conjectured that every win-lose bimatrix game has a 2/3-well-supported Nash equilibrium that uses supports of cardinality at most three. Indeed, they showed that such an equilibrium will exist subject to the correctness of a graph-theoretic conjecture. Regardless of the correctness of this conjecture, we show that the barrier of a 2/3 payoff guarantee cannot be broken with constant size supports; we construct win-lose games that require supports of cardinality at least Omega((log n)^(1/3)) in any epsilon-well supported equilibrium with epsilon < 2/3. The key tool in showing the validity of the construction is a proof of a bipartite digraph variant of the well-known Caccetta-Haggkvist conjecture. A probabilistic argument shows that there exist epsilon-well-supported equilibria with supports of cardinality O(log n/(epsilon^2)), for any epsilon> 0; thus, the polylogarithmic cardinality bound presented cannot be greatly improved. We also show that for any delta > 0, there exist win-lose games for which no pair of strategies with support sizes at most two is a (1-delta)-well-supported Nash equilibrium. In contrast, every bimatrix game with payoffs in [0,1] has a 1/2-approximate Nash equilibrium where the supports of the players have cardinality at most two.Comment: Added details on related work (footnote 7 expanded

Counting flags in triangle-free digraphs

Motivated by the Caccetta-Haggkvist Conjecture, we prove that every digraph on n vertices with minimum outdegree 0.3465n contains an oriented triangle. This improves the bound of 0.3532n of Hamburger, Haxell and Kostochka. The main new tool we use in our proof is the theory of flag algebras developed recently by Razborov.Comment: 19 pages, 7 figures; this is the final version to appear in Combinatoric

On The Graphs and Their Complements with Prescribed Circumference

Let Gt(n) be the class of connected graphs on n vertices having the longest cycle of length t and let G ∈ Gt(n). Woodall (1976) determined the maximum number of edges of G. An alternative proof and characterization of the extremal (edge-maximal) graphs given by Caccetta &amp; Vijayan (1991). The edge-maximal graphs have the property that their complements are either disconnected or have a cycle going through each vertex (i.e. they are hamiltonian). This motivates us to investigate connected graphs with prescribed circumference (length of the longest cycle) having connected complements with cycles . More specifically, we focus our investigations on the class G (n, c, c) denoting the class of connected graphs on n vertices having circumference c and whose connected complements have circumference c. The problem of interest is that of determining the bounds of the number of edges of a graph G∈ G(n, c, c) and characterize the extremal graphs of G(n, c, c). We discuss the class G (n, c, c) and present some results for small c. In particular for c=4 and c =n-2, we provide a complete solution

Two defence applications involving discrete valued optimal control

We present two examples of optimal discrete valued control problems which arise in defence applications. The first one involves the management of batteries in a submarine. In the second example, one wishes to determine an optimal transit path for a submarine through a field of sonar sensors, subject to a total time constraint. A recently developed numerical solution technique for discrete valued optimal control problems is used to solve both of these problems. We present numerical results for each example

The history of degenerate (bipartite) extremal graph problems

This paper is a survey on Extremal Graph Theory, primarily focusing on the case when one of the excluded graphs is bipartite. On one hand we give an introduction to this field and also describe many important results, methods, problems, and constructions.Comment: 97 pages, 11 figures, many problems. This is the preliminary version of our survey presented in Erdos 100. In this version 2 only a citation was complete

Phototriggered release of tetrapeptide AAPV from coumarinyl and pyrenyl cages

Ala-Ala-Pro-Val (AAPV) is a bioactive tetrapeptide that inhibits human neutrophil elastase (HNE), an enzyme involved in skin chronic inflammatory diseases like psoriasis. Caged derivatives of this peptide were prepared by proper N- and C-terminal derivatisation through a carbamate or ester linkage, respectively, with two photoactive moieties, namely 7-methoxycoumarin-2-ylmethyl and pyren-2-ylmethyl groups. These groups were chosen to assess the influence of the photosensitive group and the type of linkage in the controlled photorelease of the active molecule. The caged peptides were irradiated at selected wavelengths of irradiation (254, 300, and 350 nm), and the photolytic process was monitored by HPLC-UV. The results established the applicability of the tested photoactive groups for the release of AAPV, especially for the derivative bearing the carbamate-linked pyrenylmethyl group, which displayed the shortest irradiation times for the release at the various wavelengths of irradiation (ca. 4 min at 254 nm, 8 min at 300 nm and 46 min at 350 nm).Thanks are due to the Fundação para a Ciência e Tecnologia (FCT, Portugal) for financial support to the portuguese NMR network (PTNMR, Bruker Avance III 400- Univ. Minho), FCT and FEDER (European Fund for Regional Development)- COMPETE-QREN-EU for financial support through the Chemistry Research Centre of the University of Minho (Ref. UID/QUI/00686/2013 and UID/QUI/0686/2016). A PhD grant to A.M.S. (SFRH/BD/80813/2011) is also acknowledged.info:eu-repo/semantics/publishedVersio