Problems on graphs with few P4's and indifference graphs

Author(s)

Vagner Pedrotti

• Document type: Doctoral Thesis
• Press: Campinas, SP.
• Institution: Universidade Estadual de Campinas (UNICAMP). Instituto de Computação
• Defense date: 2011-08-19

Examining board members

Célia Picinin de Mello; Cláudia Linhares Sales; Sulamita Klein; Flávio Keidi Miyazawa; Orlando Lee

• Advisor: Célia Picinin de Mello

Abstract

In this doctoral thesis, three problems on graphs are considered and results are given for them when the input is resctricted to some graph classes. All the problems are combinatorial optimization problems on simple graphs and have distinct classihcations of complexity. In two of them, the research focused on graph classes known as graphs with "few iVs" and on the use of modular decomposition on such graphs. In the last problem, a subclass of interval graphs was studied with respect to the application of the technique known as pullback. The first problem studied is the Minimal Separator Problem. For this problem, there exists polynomial time algorithms for every class of graphs which has a polynomial number of minimal separators. A linear-time algorithm, that lists all minimal separators of extended iVladen graphs, is presented. Moreover, tight bounds on the number and on the total size of minimal separators are given for extended iVladen graphs and for some of their subclasses: the iVladen, iVtidy, and iVlite graphs. This result extends a previous algorithm for iVspai'se graphs and gives, for the above classes, better bounds on the number of minimal separators that were already known to be polynomial. Then, the Clique Packing Problem is analyzed. The problem is an extension of the classical Maximum Matching Problem and is NP-Hard for almost all graph classes. The contribution presented solves the problem in polynomial time (for any fixed clique size) in iVtidy graphs through a technique similar to that used for cographs. However, the most well-known superclasses of iVtidy graphs contains split graphs, for which this problem is NP-Hard. This is an evidence that the technique was fully explored with respect of graph classes with few iVs. At last, the Strong Total Coloring Problem is considered. It is a recently introduced variation of the classical Total Coloring Problem and its complexity is still unknown. As expected, there are quite few graph classes for which the problem has a polynomial time algorithm. Besides its complexity, another important open question for this problem is a conjecture which states that A(G) + 3 colors are sufficient for coloring any graph G. A known technique, called pullback, used for edge and total coloring of dually chordal graphs is extended to derive a linear time algorithm for indifference graphs (also known as proper interval graphs). This algorithm produces solutions that validate the conjecture for this graph class. These contributions assert the importance of modular decomposition in algorithms for graph classes with "few P4's" and broaden the pullback technique to variations of classical coloring problems (AU)