Degeneracy graphs and the neighbourhood problem

1. Introduction.- 2. The Concept of Degeneracy.- 3. Degeneracy Graphs.- 3.1 The concept of degeneracy graphs.- 3.2 Properties of degeneracy graphs.- 3.2.1 Properties of degeneracy graphs in case of simple degeneracy.- 3.2.2 Properties of degeneracy graphs in case of multiple degeneracy.- 3.3 Degeneracy tableaux.- 4. On the Number of Nodes of Degeneracy Graphs.- 4.1 The maximum number of nodes of degeneracy graphs.- 4.2 The density of degeneracy tableaux.- 4.3 The minimum number of nodes of degeneracy graphs.- 4.4 On the existence and uniqueness of ?xn-degeneracy graphs.- 4.5 An algorithm for determining the number of nodes of degeneracy graphs.- 5. A Method to Solve the Neighbourhood Problem.- 5.1 Examples of the occurrence of the neighbourhood problem.- 5.1.1 The neighbourhood problem in connection with sensitivity analysis.- 5.1.2 The neighbourhood problem in bottleneck linear programming.- 5.2 Solution of the neighbourhood problem by means of degeneracy graphs.- 5.2.1 On the existence of N-minimal trees of a positive degeneracy graph.- 5.2.1.1 The existence of N-minimal trees in case of simple degeneracy.- 5.2.1.2 The existence of quasi-N-minimal trees in case of multiple degeneracy.- 5.2.2 The N-tree method for solving the neighbourhood problem.- 5.2.2.1 The principle of the N-tree method.- 5.2.2.2 Algorithmic description of the procedure.- 5.2.2.3 The N-tree algorithm in programmable form.- 5.2.2.4 Some explanations with respect to TREE and ALL.- 5.2.3 On the efficiency of the N-tree method.- 5.2.3.1 Comparison of TREE and ALL.- 5.2.3.2 Estimations with respect to the number of nodes of TREE-solutions and N-trees.- 5.2.4 On the application of the N-tree method.- Appendices.- A. Basic concepts of linear programming and of theory of convex polytopes.- B. Basic concepts of graph theory.- C. On 2xn-degeneracy graphs.- D. Flow-charts.- References.- Index of symbols.- Index of terms.