Comparing 17 graph parameters

Abstract

Many parametrized problems were decided to be FPT or W-hard. However, there is still thousands of problems and parameters for which we do not know yet whether are FPT or W-hard. In this thesis, we provide a tool for extending existing results to additional parametrized problems. We use the comparison relation for comparing graph parameters, i.e. we say that parameter p1 is bounded by parameter p2 if exists a function f that for every graph G holds p1(G)<=f(p2(G)). This allows us to extend results for parametrized problem P in two ways. If problem P parametrized by p1 is FPT then also P parametrized by p2 is FPT. Or wise-versa, if problem P parametrized by p2 is W-hard then also P parametrized by p1 is W-hard. Moreover, we show whether is a parameter bounded by another parameter for the 17 graph parameters: path-width, tree-width, branch-width, clique-width, rank-width, boolean-width, maximum independent set, minimum dominating set, vertex cover number, maximum clique, chromatic number, maximum matching, maximum induced matching, cut-width, carving-width, degeneracy and tree-depth. To avoid all 272 different comparisons, we introduce methodology for examining graph parameters which rapidly reduce number of comparisons. And finally, we provide comparison diagram for all 17 parameters.