Wagner's theorem
In graph theory, Wagner's theorem is a mathematical forbidden graph characterization of planar graphs, named after Klaus Wagner, stating that a finite graph is planar if and only if its minors include neither K5 (the complete graph on five vertices) nor K3,3 (the utility graph, a complete bipartite graph on six vertices). This was one of the earliest results in the theory of graph minors and can be seen as a forerunner of the Robertson–Seymour theorem.
Definitions and theorem statement
A planar embedding of a given graph is a drawing of the graph in the Euclidean plane, with points for its vertices and curves for its edges, in such a way that the only intersections between pairs of edges are at a common endpoint of the two edges. A minor of a given graph is another graph formed by deleting vertices, deleting edges, and contracting edges. When an edge is contracted, its two endpoints are merged to form a single vertex. In some versions of graph minor theory the graph resulting from a contraction is simplified by removing self-loops and multiple adjacencies, while in other version multigraphs are allowed, but this variation makes no difference to Wagner's theorem. Wagner's theorem states that every graph has either a planar embedding, or a minor of one of two types, the complete graph K5 or the complete bipartite graph K3,3. (It is also possible for a single graph to have both types of minor.)
Richard Wagner
Wilhelm Richard Wagner (/ˈvɑːɡnər/; German: [ˈʁiçaʁt ˈvaːɡnɐ]; 22 May 1813 – 13 February 1883) was a German composer, theatre director, polemicist, and conductor who is primarily known for his operas (or, as some of his later works were later known, "music dramas"). Unlike most opera composers, Wagner wrote both the libretto and the music for each of his stage works. Initially establishing his reputation as a composer of works in the romantic vein of Weber and Meyerbeer, Wagner revolutionised opera through his concept of the Gesamtkunstwerk ("total work of art"), by which he sought to synthesise the poetic, visual, musical and dramatic arts, with music subsidiary to drama, and which was announced in a series of essays between 1849 and 1852. Wagner realised these ideas most fully in the first half of the four-opera cycle Der Ring des Nibelungen (The Ring of the Nibelung).
His compositions, particularly those of his later period, are notable for their complex textures, rich harmonies and orchestration, and the elaborate use of leitmotifs—musical phrases associated with individual characters, places, ideas or plot elements. His advances in musical language, such as extreme chromaticism and quickly shifting tonal centres, greatly influenced the development of classical music. His Tristan und Isolde is sometimes described as marking the start of modern music.
Podcasts:
