Extremal Paths in Graphs : Foundations, Search Strategies and Related Topics

The central problem in this book is the search for optimal paths in graphs.

The simplest example is the search for the shortest connection from one place to another one in a city.

The author investigates generalized versions of the Dijkstra algorithm and the Ford-Bellman algorithm; these generalized search strategies find paths with minimum or almost minimum costs even if the cost function is not computed by adding costs of the edges of a path.

Many sorts of optimal path problems are described, for example the search for optimal paths in random graphs or NP-complete optimal path problems like the Traveling Salesman Problem.

Also, the author studies structural properties of cost measures for paths in graphs; in particular, he investigates generalized versions of additivity, Bellman properties, and order preservation of cost functions.

Moreover, the author quotes many combinatorial results on paths in graphs.

A typical one is that the length of the longest simple cycle in an undirected graph is greater than the minimum valence of any vertex in this graph.