The spanning trees forced by the path and the star

A set S of trees of order n forces a tree T if every graph having each tree in S as a spanning tree must also have T as a spanning tree. A spanning tree forcing set for order n that forces every tree of order n. A spanning-tree forcing set S is a test set for panarboreal graphs, since a graph of order n is panarboreal if and only if it has all of the trees in S as spanning trees. For each positive integer n ≠ 1, the star belongs to every spanning tree forcing set for order n. The main results of this paper are a proof that the path belongs to every spanning-tree forcing set for each order n ∉ {1, 6, 7, 8} and a computationally tractable characterization of the trees of order n ≥ 15 forced by the path and the star. Corollaries of those results include a construction of many trees that do not belong to any minimal spanning tree forcing set for orders n ≥ 15 and a proof that the following related decision problem is NP-complete: an instance is a pair (G, T) consisting of a graph G of order n and maximum degree n - 1 with a hamiltonian path, and a tree T of order n; the problem is to determine whether T is a spanning tree of G. © 1996 John Wiley & Sons, Inc.     

A TEM and XRD study of the SbNbSe3 misfit layer structures

Two new misfit layer structures have been synthesized within the Sb-Nb-Se system. Powder X-ray diffraction and electron microscopy techniques (electron diffraction, HREM, XEDS) have been used to determine the nature of their structure. According to TEM and XEDS data (for more than 15 crystals studied) both phases are monolayer type, i.e. (SbSe)1+delta (NbSe2). Electron microscopy reveals a composite modulated structure that consists of the periodical intergrowth of a pseudotetragonal SbSe layer, denominated as Q, and a pseudohexagonal layer NbSe2, denominated as H. Both layers fit along b, stack along c and do not fit along a (misfit) giving rise to an incommensurate modulation along this direction. The two phases differ in the symmetry of the Q layers being in one case orthorhombic (for delta = 0.17) and monoclinic in the other (for delta = 0.19). After the characterization of the sample by electron microscopy the unit cells of the basic layers could be refined for both phases by powder X-ray diffraction: aQ = 5.824(2) A, bQ = 5.962(5) A, cQ = 23.927(6) A, alpha = 90 degrees, beta = 90 degrees and gamma = 90 degrees and aH = 3.415(5) A, bH = 5.962(6) A,, cH = 11.962(1) A, alpha = 90 degrees, beta = 90 degrees and gamma = 90 degrees for the orthorhombic phase; aQ = 5.844(2) A, bQ = 5.981(1) A, cQ = 23.919(5) A, alpha = 90 degrees, beta = 90 degrees and gamma = 96.00(3)degrees and aH = 3.439(1) A, bH = 5.994(2) A, cH = 11.956(3) A, alpha = 90 degrees, beta = 90 degrees and gamma = 90 degrees for the monoclinic phase. The phase with the monoclinic Q-sublattice often appears as twinned crystals. The more abundant crystals are disordered intergrowths of both monolayer phases.