Generalized vertex coloring problems using split graphs

Graph theory experienced a remarkable increase of interest among the scientific community during the last decades. The vertex coloring problem (Min Coloring) deserves a particular attention rince it has been able to capture a wide variety of applications. For mathematicians, it is interesting for an additional reason: it is extremely hard to solve it in an efficient way. In this thesis, we introduce several problems generalizing the usual vertex coloring problem, and hence, extending its application domain. We say that a graph is (p, k)-colorable if its vertex set can be partitioned into p cliques and k stable sets. Then, for a given p (respectively k), one may ask the following questions: how to choose p cliques (respectively k stable sets) to be removed from the graph such that the number of stable sets (respectively cliques) partitioning the remaining vertices is minimized? These are called (p, k)-coloring problems. We also introduce Min Split-coloring which is, given a graph G, the problem of minimizing k such that G is (k, k)-colorable. Along the saine line, given a graph G, the objective of the problem Min Cocoloring is to minimize p + k such that G is (p, k)-colorable. All these problems, called together generalized coloring problems, are obviously at least as difficult as Min Coloring. The purpose of this dissertation is to study generalized coloring problems in nome restricted classes of graphs in order to bring a new insight on the relative difficulties of these problems. To this end, we detect in a more precise way the limits between NP-hard and polynomially solvable problems. Chapter 1 introduces generalized coloring problems by emphasizing nome preliminary results which will guide the questions to handle in the following chapters. Chapter 2 exposes the first clans of graphs, namely cacti, where Min Split-coloring is shown to be polynomially solvable. We also observe that generalized coloring problems can be polynomially solved in triangulated graphs. The main result of Chapter 3 is a new characterization of cographs: it is equivalent to say that G is a cograph, and to state that, for every subgraph G' ⊆ G, G' is (p, k)-colorable if and only if G' [V \ K] is (p – 1, k)-colorable, where K induces a maximum clique of G'. This result implies simple combinatorial algorithme to solve all generalized coloring problems; the one for Min Cocoloring improves the best time complexity known so far. In Chapter 4, we handle the recognition of polar graphs which can be seen as a particular (p, k)-coloring, where p cliques are independent (i.e., not linked at all) and k stable sets form a complete k-partite graph. It is known that the recognition of polar graphs is NP-complete. Here, we determine the first clans of graphs, namely cographs, where the polar graphs can be recognized in polynomial time, more precisely in time O(n log n). We also give a characterization by forbidden subgraphs. In the came manner, we characterize monopolar cographs, i.e., cographs admitting a polar partition with at most one clique or at most one stable set. Chapter 5 is devoted to generalized coloring problems in line graphs. Here, we detect the first classes of graphs, namely line graphs of trees, line graphs of bipartite graphs and line graphs of line-perfect graphs, where generalized coloring problems diverge in terms of NP-hardness. In Chapter 6 we study the approximability of generalized coloring problems in line graphs, in comparability graphs and in general graphs. We derive approximation algorithms with a performance guarantee using both the standard approximation ratio and the differential approximation ratio. We show that both Min Split-coloring and Min Cocoloring are at least as hard as Min Coloring to approximate from the standard approximation ratio point of view, whereas, they admit a polynomial time differential approximation scheme and Min Coloring only a constant differential approximation ratio. We also show that Min Cocoloring reduces to Min Split-coloring in all classes of graphs closed under addition of disjoint cliques and under join of a complete k-partite graph. In Chapter 7, we handle two different applications of Min Split-coloring in permutation graphs. They give birth to a new problem, called Min Threshold-coloring, that we study in the came spirit as the other generalized coloring problems. In the last chapter, we present several open questions arising from this thesis.