
StatusThe thesis was presented on the 23 December, 2008Approved by NCAA on the 26 February, 2009 Abstract– 0.26 Mb / in romanianThesisCZU 519.17
0.85 Mb /
in romanian 
The PhD thesis is dedicated of study of dconvexity on discrete structures through undirected and directed graphs.
There is solved the problem of characterisation of the set of dconvex simple and undirected graphs, that begins in 6070 years of the last century. These graphs are described by using an iterative method. There are introduced a sequence of algebraic operations on the set of dconvex simple undirected graphs, that allows so to extend some known classes of graphs, also finding of some primary graphs, from which, by using these operation, could be formed the new classes of dconvex simple graphs. By this way it is determined the structure of undirected dconvex simple graphs, that is important for substantiation of convexity theory on discrete structures and for applications. It is proved the theorem of existence of bipartite dconvex simple graphs with fixed number of vertexes and edges. It is shown that the metric problems, in dconvex simple graph of type L(G, G0) are reducible to the same problems in graph G. It is determined the dimension of involving in Euclidian spaces of planar dconvex simple graphs and it is described algorithm for involving of these graphs in these spaces.
Like in case of the dconvex simple graphs, the dconvex quasisimple graphs are described by using an iterative method. There are defined all operations that where defined for undirected d convex simple graphs. By using them, there are studied the set of dconvex quasisimple graphs by splitting in some special classes and are found generators for some of them. There is determined the dimension of involving in Euclidian spaces of dconvex quasisimple planar graphs and described algorithm for involving of these graphs in these spaces.
By using the notion of distance, defined in an usual way on the set of the directed graphs, which is a semimetric of these set of graphs, there is defined the notion of dconvex set in directed graphs.
This notion is in concordance with the axiomatic theory of convexity. There are defined the directed dconvex simple graphs, as graphs with a minimal number of the dconvex sets. The dconvex simple directed graphs are described by using an iterative method. The operations, which were defined for dconvex simple and dconvex quasisimple undirected graphs, in addition with the operation of transposition and the Loperation, are defined on the set of directed dconvex simple graphs. It is proved that these operations are algebraic on the set of directed dconvex simple graphs, too. All this are necessary so for emphasize the structure of these graphs and the way of them constructions, also for to show that for each undirected dconvex simple graph, with known structure, there is at least one directed dconvex simple graphs, that correspond to the first, and correspondence is searched in the set of directions(orientations) of the initial undirected graph. So, it could be assert that the set of directed dconvex simple graphs contains, in this mean, the set of known undirected dconvex simple set.