Partitioning extended [formula omitted]-laden graphs into cliques and stable sets

Two graphs G and H with the same vertex set V are P4-isomorphic if every four vertices {a, b, c, d} ⊆ V induce a chordless path (denoted by P4) in G if and only if they induce a P4 in H.We call a graph split-perfect if it is P4-isomorphic to a split graph (i.e. a graph being partitionable into a clique and a stable set). This paper characterizes the new class of split-perfect graphs using the concepts of homogeneous sets and p-connected graphs, and leads to a linear time recognition algorithm for split-perfect graphs, as well as linear time algorithms for classical optimization problems on split-perfect graphs based on the primeval decomposition of graphs. These results considerably extend previous ones on smaller classes such as P4-sparse graphs, P4-lite graphs, P4-laden graphs, and (7,3)-graphs. Moreover, split-perfect graphs form a new subclass of brittle graphs containing the superbrittle graphs for which a new characterization is obtained leading to linear time recognition.

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.

Set covering and packing formulations of graph coloring: Algorithms and first polyhedral results

Partitioning <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e146" altimg="si286.svg"><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub></mml:math>-tidy graphs into a stable set and a forest

A note on the computational complexity of graph vertex partition

The median of a set of vertices P of a graph G is the set of all vertices x of G minimizing the sum of distances from x to all vertices of P. In this paper, we present a linear time algorithm to compute medians in median graphs, improving over the existing quadratic time algorithm. We also present a linear time algorithm to compute medians in the 𝓁₁-cube complexes associated with median graphs. Median graphs constitute the principal class of graphs investigated in metric graph theory and have a rich geometric and combinatorial structure. Our algorithm is based on the majority rule characterization of medians in median graphs and on a fast computation of parallelism classes of edges (Θ-classes or hyperplanes) via Lexicographic Breadth First Search (LexBFS). To prove the correctness of our algorithm, we show that any LexBFS ordering of the vertices of G satisfies the following fellow traveler property of independent interest: the parents of any two adjacent vertices of G are also adjacent.

A stable set in a graph G is a set of pairwise nonadjacent vertices. The problem of finding a maximum weight stable set is one of the most basic ℕℙ-hard problems. An important approach to this problem is to formulate it as the problem of optimizing a linear function over the convex hull STAB(G) of incidence vectors of stable sets. Since it is impossible (unless ℕℙ=coℕℙ) to obtain a “concise” characterization of STAB(G) as the solution set of a system of linear inequalities, it is a more realistic goal to find large classes of valid inequalities with the property that the corresponding separation problem (given a point x *, find, if possible, an inequality in the class that x * violates) is efficiently solvable.¶Some known large classes of separable inequalities are the trivial, edge, cycle and wheel inequalities. In this paper, we give a polynomial time separation algorithm for the (t)-antiweb inequalities of Trotter. We then introduce an even larger class (in fact, a sequence of classes) of valid inequalities, called (t)-antiweb-s-wheel inequalities. This class is a common generalization of the (t)-antiweb inequalities and the wheel inequalities. We also give efficient separation algorithms for them.

A unified approach to recognize squares of split graphs

Probe interval and probe unit interval graphs on superclasses of cographs

Graph Theory A graph is probe (unit) interval if its vertices can be partitioned into two sets: a set of probe vertices and a set of nonprobe vertices, so that the set of nonprobe vertices is a stable set and it is possible to obtain a (unit) interval graph by adding edges with both endpoints in the set of nonprobe vertices. Probe (unit) interval graphs form a superclass of (unit) interval graphs. Probe interval graphs were introduced by Zhang for an application concerning the physical mapping of DNA in the human genome project. The main results of this article are minimal forbidden induced subgraphs characterizations of probe interval and probe unit interval graphs within two superclasses of cographs: P4-tidy graphs and tree-cographs. Furthermore, we introduce the concept of graphs class with a companion which allows to describe all the minimally non-(probe G) graphs with disconnected complement for every graph class G with a companion.

Bisplit graphs

In this chapter we survey the results of the polyhedral approach to a particular NP-hard combinatorial optimization problem, the stable set problem in graphs. (Alternative names for this problem used in the literature are vertex packing, or coclique, or independent set problem.) Our basic technique will be to look for various classes of inequalities valid for the stable set polytope, and then develop polynomial time algorithms to check if a given vector satisfies all these constraints. Such an algorithm solves a relaxation of the stable set problem in polynomial time, i. e., provides an upper bound for the maximum weight of a stable set. If certain graphs have the property that every facet of the stable set polytope occurs in the given family of valid inequalities, then, for these graphs, the stable set problem can be solved in polynomial time. It turns out that there are very interesting classes of graphs which are in fact characterized by such a condition, most notably the class of perfect graphs. Using this approach, we shall develop a polynomial time algorithm for the stable set problem for perfect graphs. So far no purely combinatorial algorithm has been found to solve this problem in polynomial time.KeywordsPolynomial TimeBipartite GraphValid InequalityInterval GraphPerfect GraphThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Separation of Antiweb-Wheel Inequalities Over Stable Set Polytopes

A proper coloring of a graph is a partition of its vertex set into stable sets, where each part corresponds to a color. For a vertex-weighted graph, the weight of a color is the maximum weight of its vertices. The weight of a coloring is the sum of the weights of its colors. Guan and Zhu defined the weighted chromatic number of a vertex-weighted graph G as the smallest weight of a proper coloring of G (1997). If vertices of a graph have weight 1, its weighted chromatic number coincides with its chromatic number. Thus, the problem of computing the weighted chromatic number, a.k.a. Max Coloring Problem, is NP-hard in general graphs. It remains NP-hard in some graph classes as bipartite graphs. Approximation algorithms have been designed in several graph classes, in particular, there exists a PTAS for trees. Surprisingly, the time-complexity of computing this parameter in trees is still open. The Exponential Time Hypothesis (ETH) states that 3-SAT cannot be solved in sub-exponential time. We show that, assuming ETH, the best algorithm to compute the weighted chromatic number of n-node trees has time-complexity n (log n). Our result mainly relies on proving that, when computing an optimal proper weighted coloring of a graph G, it is hard to combine colorings of its connected components.

A proper coloring of a graph is a partition of its vertex set into stable sets, where each part corresponds to a color. For a vertex-weighted graph, the weight of a color is the maximum weight of its vertices. The weight of a coloring is the sum of the weights of its colors. Guan and Zhu defined the weighted chromatic number of a vertex-weighted graph $G$ as the smallest weight of a proper coloring of $G$. If vertices of a graph have weight 1, its weighted chromatic number coincides with its chromatic number. Thus, the problem of computing the weighted chromatic number, a.k.a. the max coloring problem, is NP-hard in general graphs. It remains NP-hard in some graph classes as bipartite graphs. Approximation algorithms have been designed in several graph classes; in particular, there exists a polynomial-time approximation scheme for trees. Surprisingly, the time-complexity of computing this parameter in trees is still open. The exponential time hypothesis (ETH) states that 3-SAT cannot be solved in subexpon...