IA Scholar Query: Farkas certificates and minimal witnesses for probabilistic reachability constraints.
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Internet Archive Scholar query results feedeninfo@archive.orgThu, 30 Dec 2021 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Cycle structure and colorings of directed graphs
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This thesis deals with problems from the theory of finite directed graphs. A directed graph (digraph for short) is a binary relation whose domain has finite size. With that digraphs can be seen as a very general way of representing (possibly asymmetric) relations between pairs from a finite set of objects. Undoubtedly, such a generality allows to encode many structures by digraphs. This works particularly well if important properties of the structure at hand can be expressed as relations or connections between objects. To give some selected examples, let us mention road networks, electricity networks, radio networks, the world wide web, circuits in electronic devices, or neural networks. A main focus of the thesis at hand is the investigation of properties of one of the most fundamental objects all over graph theory, the so-called cycle (sometimes also called circuit). A cycle in a graph is determined by a closed alternating sequence of cyclically connected vertices and edges. In a graph of finite size one will typically see loads of distinct cycles of various types. Therefore cycles constitute an important and recurring motive in almost all branches of graph theory, for instance, they play important roles in structural graph theory, in the theory of flows on directed networks, in theoretical characterizations of graph classes, as well as in the theory of graph colorings. Additionally, cycles play a decisive role in numerous algorithmic problems and their solutions, such as in the Traveling Salesman Problem, algorithms for finding a largest matching in a given graph, in the max-flow problem, and also in subprocedures such as Kruskal's algorithm for finding a minimum weight spanning tree. For those reasons, a substantial amount of research in graph theory has specialised on the structure of cycles in graphs. In the first major part of this thesis we deal with cycles which occur in directed graphs, and prove several necessary and sufficient theoretical conditions for the existence of cycles of certain types. Additi [...]Raphael Mario Steiner, Technische Universität Berlin, Stefan Felsnerwork_4oljppo7pnh3vlrxkkkho6zd4mThu, 30 Dec 2021 00:00:00 GMT