We give a complete answer to a question raised by Harary and Manvel in 1972 [3] by proving that a finite set A of points in the plane R is uniquely determined up to translation and rotation by a multiple of 90◦ by 5 of its (|A| − 1)-element subsets given up to translation and rotation by a multiple of 90◦. This result is best-possible.

We study 0-1 laws for extensions of rst-order logic by Lindstrr om quantiiers. We state suucient conditions on a quantiier Q expressing a graph property, for the logic FOQ] { the extension of rst-order logic by means of the quantiier Q { to have a 0-1 law. We use these conditions to show, in particular, that FORig], where Rig is the quantiier expressing rigidity, has a 0-1 law. We also show tha...

Harary et al. put forward the concept on minimum cardinality over all subsets of perfect matching M that are not included by any other ones, to be forcing number for M. A counting polynomial matchings possessing same was introduced Zhang al., using name `forcing polynomial'. This research deduces recurrence formulas polynomials monotonic constructable hexagonal systems and with one turning. Fro...

A linear forest is a in which every connected component path. The arboricity of graph G the minimum number forests covering all edges. In 1980, Akiyama, Exoo, and Harary proposed conjecture, known as Linear Arboricity Conjecture (LAC), stating that Δ-regular has ⌈ Δ + 1 2 ⌉ . 1988, Alon proved LAC holds asymptotically. 1999, list version was raised by An Wu, called List Conjecture. this article...

CONTEXT Atrial fibrillation is a common, but potentially preventable, complication following coronary artery bypass graft (CABG) surgery. OBJECTIVES To assess the nature and consequences of atrial fibrillation after CABG surgery and to develop a comprehensive risk index that can better identify patients at risk for atrial fibrillation. DESIGN, SETTING, AND PARTICIPANTS Prospective observati...

A graph H is an edge reconstruction of the graph G if there is a bijection p from E(G) to E(H) such that for each edge e in E(G), G\e is isomorphic to EI\fi(e). (Here G\e denotes the graph obtained from G by deleting the edge e.) We call G edge reconstructible if any edge reconstruction of G is isomorphic to G. The well-known edge-reconstruction conjecture, due to Harary [4], asserts that any g...