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Let G = (V, E) be a nontrivial connected graph. For a subset S ⊆ V, the geodesic closure (S) of S is the set of all vertices on geodesics (shortest paths) between two vertices of S. We study the geodetic achievement and avoidance games defined by Buckley and Harary (Geodetic games for graphs, Quaestiones Math. 8 (1986), 321–334) as follows. The first player A chooses a vertex v1 of G. The second player B then selects v2 ≠ v1 and determines the geodetic closure (S 2) for S 2 = {v 1 , v 2 }. If (S 2) = V, then the second player wins the achievement game, but loses the avoidance game. If (S 2) = V, then A picks v 3 ∉ S 2 and determines (S 3) for S 3 = {v 1 , v 2 , v 3 }. In general, A and B alternatively select a new vertex in this manner. The first player who selects a vertex v k such that (S k) = V wins the achievement game; in the avoidance game he is the loser. We solve these games for several families of graphs, including trees and complete multipartite graphs, by determining which player is the winner.

For a vertex x of G, the eccentricity e (x) is the distance between x and a vertex farthest from x. Then x is a contour vertex if there is no neighbor of x with its eccentricity greater than e (x). The x-y path of length d (x,y) is called a x-y geodesic. The geodetic interval I [x,y] of a graph G is the set of vertices of all x-y geodesics in G. For S ⊆ V , the geodetic closure I [S] of S is the union of all geodetic intervals I [x,y] over all pairs x,y ∈S. A vertex set S is a geodetic set for G if I [S] = V (G). In this thesis, we study the contour sets of product graphs and discuss these sets are geodetic sets for some conditions.

This thesis examines the effect of spatial resolution on lithological and alteration mapping using remotely sensed multispectral data. The remotely sensed data were obtained by the Thematic Mapper (TM) and Airborne Thematic Mapper (ATM) over two areas in the Arabian Shield. These were the Mahd Adh Dhahab and Jabal Said areas. The ATM data had a nominal spatial resolution of 7.5m, 5m, and 2.5m. In order to compare these data sets it was necessary to correct for, sensor- and scene-related distortions. This was achieved by calibrating each data set and converting them to reflectance units using ground spectra with a similar spectral resolution obtained with the Barringer Hand Held Ratioing Radiometer (HHRR) . The ATM data were also corrected for X-track shading by normalising the brightness of each column to that of the centre column. The result of X-ray and laboratory spectral analysis of samples collected from the study areas, support the presence of characteristic minerals associated with the alteration zones. The corrected data were analysed by a variety of techniques in order to enhance the geological information present in the data. These included false colour compositing, decorrelating stretching and band ratioing. The latter two techniques proved most effective for discrimination and several additional geological units and areas were identified which had not been mapped previously. Results further indicate that the increased spatial resolution of the ATM data did not permit greater discrimination than the TM data. This suggests TM data should prove to be a cost-effective way of mapping and detection of alteration zones in the Arabian Shield.

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Bibliography: pages 113-117. / The Department of Surveying and Geodetic Engineering at the University of Cape Town, in conjunction with the Department of Mechanical Engineering at the University of Cape Town have developed a non-contact co-ordinate measurement machine in a project called MILIMAP. The project had the following objectives : 1. To determine unique surface co-ordinates for continuous, complex objects with submillimetre accuracy. 2. The representation of the co-ordinates was to be in a format that could be utilised by a computer numerically controlled (CNC) milling machine in a computer aided design/ computer aided manufacture (CADCAM) environment. 3. The device had to use a non-contact method for data capture. The MILIMAP project was undertaken because there is a demand for co-ordinate measurement machines in industry for the inspection of objects for quality control purposes. Conventional Co-ordinate Measurement Machines (CMMs) are expensive and use a contact probe to measure the object. The contact probe measurement technique is unsuitable for the measurement of non-rigid objects such as shoes and automobile seat padding. The MILIMAP system provides a noncontact measurement technique that can be applied to non-rigid as well as rigid objects. Additional applications in the archaeological field exist for the non-contact measurement of sensitive, historical artefacts. A digital photogrammetric system was developed to measure the position of a laser dot projected onto the surface of the measurement object. This measurement system satisfied the criteria of a non-contact measurement method required for the project. The system utilised three digital CCD cameras to capture images of the laser dot projected onto the object. Image processing software, developed from existing software within the Department of Surveying and Geodetic Engineering, was used to photogrammetrically determine the co-ordinates of the laser dot to sub-millimetre accuracy on the surface of the object. A mechanical device was designed and constructed by the Department of Mechanical Engineering in order to move the laser over the surface of the object, and to rotate the object. The entire surface of the object could be measured by the system using these operations.