In this paper, we continue the study of domination and total domination in cubic graphs. It is known [Henning M.A., Southey J., A note on graphs with disjoint dominating and total dominating sets, Ars Combin., 2008, 89, 159–162] that every cubic graph has a dominating set and a total dominating set which are disjoint. In this paper we show that every connected cubic graph on nvertices has a total dominating set whose complement contains a dominating set such that the cardinality of the total dominating set is at most (n+2)/2, and this bound is essentially best possible.
A dominating set of a graph is a set of vertices such that every vertex not in the set is adjacent to a vertex in the set, while a paired-dominating set of a graph is a dominating set such that the subgraph induced by the dominating set contains a perfect matching. In this paper, we show that no minimum degree is sufficient to guarantee the existence of a disjoint dominating set and a paired-dominating set. However, we prove that the vertex set of every cubic graph can be partitioned into a dominating set and a paired-dominating set.
Electronic autocollimators are utilised versatilely for non-contact angle measurements in applications like straightness measurements and profilometry. Yet, no calibration of the angle measurement of an autocollimator has been available when both its measurement axes are engaged. Additionally, autocollimators have been calibrated at fixed distances to the reflector, although its distance may vary during the use of an autocollimator. To extend the calibration capabilities of the Physikalisch-Technische Bundesanstalt (PTB) regarding spatial angles and variable distances, a novel calibration device has been set up: the spatial angle autocollimator calibrator (SAAC). In this paper, its concept and its mechanical realisation will be presented. The focus will be on the system’s mathematical modelling and its application in spatial angle calibrations. The model considers the misalignments of the SAAC’s components, including the non-orthogonalities of the measurement axes of the autocollimators and of the rotational axes of the tilting unit. It allows us to derive specific measurement procedures to determine the misalignments in situ and, in turn, to correct the measurements of the autocollimators. Finally, the realisation and the results of a traceable spatial angle calibration of an autocollimator will be presented. This is the first calibration of this type worldwide.
Autocollimators are versatile devices for the contactless measurement of the tilt angles of reflecting surfaces. In their practical application, e.g., in deflectometric profilometers for the precision form measurement of optical surfaces, the autocollimator beam is deflected in two orthogonal angular directions simultaneously. The concurrent engagement of both measuring axes results in errors in their angle response due to the crosstalk between them, which need to be calibrated. In this contribution, the capabilities of autocollimator calibration at the Physikalisch-Technische Bundesanstalt (PTB) are presented. The measurement of spatial angles is discussed in detail with a focus on achieving traceability of this measurand and reaching lowest uncertainties. A novel concept is introduced, which makes use of an innovative Cartesian arrangement of three autocollimators (two reference autocollimators and the autocollimator to be calibrated) facing a reflector cube. Each of the two reference autocollimators, which are used for the precise measurement of the cube’s angular orientation, is primarily sensitive to rotations of the cube around one of the two relevant axes and can, thus, be calibrated and traced back to PTB’s national primary standard for the plane angle in a conventional manner. In this way, the measurement of spatial angles is effectively divided into two separate measurements of plane angles. The mechanical realisation of the setup at PTB is described.