Title:

The complexity of greedy algorithms on ordered graphs

Let p be any fixed polynomial time testable, nontrivial, hereditary property of graphs. Suppose that the vertices of a graph G are not necessarily linearly ordered but partially ordered, where we think of this partial order as a collection of (possibly exponentially many) linear orders in the natural way. In the first part of this thesis, we prove that the problem of deciding whether a lexicographically first maximal (with respect to one of these linear orders) subgraph of G satisfying p, contains a specified vertex is NPcomplete. For some of these properties p we then show that by applying certain restrictions the problem still remains NPcomplete, and show how the problem can be solved in deterministic polynomial time if the restrictions imposed become more severe. Let H be a fixed undirected graph. An Hcolouring of an undirected graph G is a homomorphism from G to H. In the second part of the thesis, we show that, if the vertices of G are partially ordered then the complexity of deciding whether a given vertex of G is in a lexicographically first maximal Hcolourable subgraph of G is NPcomplete, if H is bipartite, and Sp2complete, if H is nonbipartite. We then show that if the vertices of G are linearly, as opposed to partially, ordered then the complexity of deciding whether a given vertex of G is in the lexicographically first maximal Hcolourable subgraph of G is Pcomplete, if H is bipartite, and DP2complete, if H is nonbipartite. In the final part of the thesis we show that the results obtained can be paralleled in the setting of graphs where orders are given by degrees of the vertices.
