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On Some Covering, Partition and Connectivity Problems in Graphs


Issac,  Davis
Algorithms and Complexity, MPI for Informatics, Max Planck Society;
International Max Planck Research School, MPI for Informatics, Max Planck Society;

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Issac, D. (2019). On Some Covering, Partition and Connectivity Problems in Graphs. PhD Thesis, Universität des Saarlandes, Saarbrücken. doi:10.22028/D291-29620.

Cite as: https://hdl.handle.net/21.11116/0000-0004-D665-9
We look at some graph problems related to covering, partition, and connectivity. First, we study the problems of covering and partitioning edges with bicliques, especially from the viewpoint of parameterized complexity. For the partition problem, we develop much more efficient algorithms than the ones previously known. In contrast, for the cover problem, our lower bounds show that the known algorithms are probably optimal. Next, we move on to graph coloring, which is probably the most extensively studied partition problem in graphs. Hadwiger’s conjecture is a long-standing open problem related to vertex coloring. We prove the conjecture for a special class of graphs, namely squares of 2-trees, and show that square graphs are important in connection with Hadwiger’s conjecture. Then, we study a coloring problem that has been emerging recently, called rainbow coloring. This problem lies in the intersection of coloring and connectivity. We study different variants of rainbow coloring and present bounds and complexity results on them. Finally, we move on to another parameter related to connectivity called spanning tree congestion (STC). We give tight bounds for STC in general graphs and random graphs. While proving the results on