Christian Smart

Studies of Graph Based LP/ILP Parameters

LP graph theory, which involves the application of linear programming and linear algebra to graph theory, is a relatively new and comparatively little studied area within graph theory. Graph theoretic parameters which have linear programming formulations are the focus of study.

Closed neighborhood order parameters are studied. Complexity results, applications of the Automorphism Class Theorem, and results involving the interrelationship of four of the closed neighborhood order parameters are presented.

A new parameter, domination-coverage, is introduced. Bounds, examples, and complexity results for this parameter are presented.

The total matrix of a graph is introduced and is used to relate a number of diverse parameters, including the independent domination, upper covering, and independent redundance. The total matrix, which arose as the constraint matrix of the integer programming formulation of independent domination, also yields new graph theoretic parameters. Two of these, independent domination-coverage and vertex-edge packing, are studied in detail. Bounds, examples, and NP-completeness results for these two parameters are presented.