Final Report Summary - DASTCO (Developing and Applying Structural Techniques for Combinatorial Objects)
Graphs are mathematical models of the many real-world networks which surround us such as the network of friendships on social media or the interactions between a collection of proteins in biology. Structural graph theory seeks to identify all the graphs with a given desirable property by decomposing the graphs into smaller graphs each contained in some basic class. The structural approach has proven particularly useful in the field in proving theoretical properties of graphs as well as in the development of efficient algorithms on graphs.
ERC Project DASTCO developed structural tools and techniques for labeled graphs, graphs with additional information assigned to the vertices and edges. Labeled graphs generalize a variety of well studied classes of graphs such as directed graphs, group labeled graphs, and signed graphs. The project has led the development of structural techniques for group labeled graphs, including the proof of a broad generalization to group labeled graphs of a classic result of Erdos and Posa on disjoint cycles in graphs which implies a number of previously studied variants of the problem. Additionally, Project DASTCO did foundational work in the development of a structural theory for directed graphs and excluded minors in directed graphs. We have shown an exact characterization of the sets of directed graphs which are well-quasi-ordered under directed minors, and we have found an efficient algorithm for the directed disjoint paths problem with congestion two in highly connected directed graphs.
The deeper understanding of graphs resulting from the study of labeled graphs has yielded further insight into classical problems of graph theory, allowing us to resolve a number of open questions in the field. The graph minor structure theorem of Robertson and Seymour has had a profound impact in classical graph theory, but the original proof runs over 400 pages does not give bounds on the parameters in the statement. Under Project DASTCO, we have discovered a simpler proof with explicit single-exponential bounds on the parameters, answering a challenge of Lovasz from 2005 to find such a proof. Separate work led to the resolution of an open question of Robertson and Seymour from 1986 on a generalization of Erdos-Posa’s cycle packing theorem to subdivisions in graphs.
ERC Project DASTCO developed structural tools and techniques for labeled graphs, graphs with additional information assigned to the vertices and edges. Labeled graphs generalize a variety of well studied classes of graphs such as directed graphs, group labeled graphs, and signed graphs. The project has led the development of structural techniques for group labeled graphs, including the proof of a broad generalization to group labeled graphs of a classic result of Erdos and Posa on disjoint cycles in graphs which implies a number of previously studied variants of the problem. Additionally, Project DASTCO did foundational work in the development of a structural theory for directed graphs and excluded minors in directed graphs. We have shown an exact characterization of the sets of directed graphs which are well-quasi-ordered under directed minors, and we have found an efficient algorithm for the directed disjoint paths problem with congestion two in highly connected directed graphs.
The deeper understanding of graphs resulting from the study of labeled graphs has yielded further insight into classical problems of graph theory, allowing us to resolve a number of open questions in the field. The graph minor structure theorem of Robertson and Seymour has had a profound impact in classical graph theory, but the original proof runs over 400 pages does not give bounds on the parameters in the statement. Under Project DASTCO, we have discovered a simpler proof with explicit single-exponential bounds on the parameters, answering a challenge of Lovasz from 2005 to find such a proof. Separate work led to the resolution of an open question of Robertson and Seymour from 1986 on a generalization of Erdos-Posa’s cycle packing theorem to subdivisions in graphs.