Loránd University Institute of Mathematics Ph . D . thesis Connectivity augmentation algorithms László Végh

The main subject of the thesis is connectivity augmentation: we would like to make a given graphk-connected by adding a minimum number of new edges. There are four basic problems in thisfield, since one might consider both edgeand node-connectivity augmentation in both graphsand digraphs. The thesis wishes to contribute to three out of these four problems: directed-and undirected node-connectivity and undirected edge-connectivity augmentation. Althoughdirected edge-connectivity augmentation is not being considered, the last chapter is devoted toa constructive characterization result related to directed edge-connectivity. Let us summarizethe main results of the thesis. • We present a min-max formula and a combinatorial polynomial time algorithm for aug-menting undirected node-connectivity by one. The complexity status of undirected node-connectivity augmentation of arbitrary graphs is still open; already the special case ofaugmenting by one has attracted considerable attention. The formula proved in Chap-ter 3 was conjectured by Frank and Jordán in 1994. • We present the first combinatorial polynomial time algorithm for directed node-connec-tivity augmentation. For this problem, Frank and Jordán gave a min-max formula in1995; however, it remained an open problem to develop a combinatorial algorithm. Wepresent two, completely different combinatorial algorithms. Chapter 2 contains one forthe special case of augmenting connectivity by one (a joint work with András Frank), andChapter 4 presents another for augmenting the connectivity of arbitrary digraphs (a jointwork with András Benczúr Jr.). The latter result also gives a new, algorithmic proof ofthe general theorem of Frank and Jordán on covering positively crossing supermodularfunctions on set pairs. • We establish a constructive characterization of (k, l)-edge-connected digraphs. This resultof Chapter 6, a joint work with Erika Renáta Kovács, settles a conjecture of Frank from2003. The theorem gives a common generalization of a number of previously known char-acterizations, and naturally fits into the framework defined by splitting off and orientationtheorems. • We present partial results concerning partition constrained undirected local edge-conn-ectivity augmentation. In Chapter 5, we discuss some classical results concerning undi-rected edge-connectivity augmentation in a unified framework, based on the technique ofedge-flippings. For the partition constrained problem we formulate a conjecture and givea partial proof. Most results are based on the papers [36], [74], [73] and [56], except for Chapter 5, whichcontains unpublished results.