Weighted stability number of graphs and weighted satisfiability: The two facets of pseudo-Boolean optimization

We exhibit links between pseudo-Boolean optimization, graph theory and logic. We show the equivalence of maximizing a pseudo-Boolean function and finding a maximum weight stable set; symmetrically minimizing a pseudo-Boolean function is shown to be equivalent to solving a weighted satisfiability proble

On Split-Coloring Problems

We study a new coloring concept which generalizes the classical vertex coloring problem in a graph by extending the notion of stable sets to split graphs. First of all, we propose the packing problem of finding the split graph of maximum size where a split graph is a graph G = (V,E) in which the vertex set V can be partitioned into a clique K and a stable set S. No condition is imposed on the edges linking vertices in S to the vertices in K. This maximum split graph problem gives rise to an associated partitioning problem that we call the split-coloring problem. Given a graph, the objective is to cover all his vertices by a least number of split graphs. Definitions related to this new problem are introduced. We mention some polynomially solvable cases and describe open questions on this are

A constrained sports scheduling problem

AbstractA real case of sports scheduling problem is presented. A calendar for two leagues has to be constructed; besides the usual restrictions on the alternation of home- and away-games, one has to consider the fact that some pairs of teams in the two leagues share the same facilities and cannot play home-games simultaneously. Furthermore depending on the results in the first games, one of the leagues is divided for the last games into two subleagues. An optimal solution is constructed by using properties of oriented factorizations of complete graphs

Stability in CAN-free graphs

AbstractA class F of graphs characterized by three forbidden subgraphs C, A, N is considered; C is the claw (the unique graph with degree sequence (1, 1, 1, 3)), A is the antenna (a graph with degree sequence (1, 2, 2, 3, 3, 3) which does not contain C), and N is the net (the unique graph with degree sequence (1, 1, 1, 3, 3, 3)). These graphs are called CAN-free. A construction is described which associates with every CAN-free graph G another CAN-free graph G′ with strictly fewer nodes than G and with stbility number α(G′) = α(G) − 1. This gives a good algorithm for determining the stability number of CAN-free graphs

Feasible edge colorings of trees with cardinality constraints

AbstractA variation of preemptive open shop scheduling corresponds to finding a feasible edge coloring in a bipartite multigraph with some requirements on the size of the different color classes. We show that for trees with fixed maximum degree, one can find in polynomial time an edge k-coloring where for i=1,…,k the number of edges of color i is exactly a given number hi, and each edge e gets its color from a set ϕ(e) of feasible colors, if such a coloring exists. This problem is NP-complete for general bipartite multigraphs. Applications to open shop problems with costs for using colors are described

Graph coloring with cardinality constraints on the neighborhoods

AbstractExtensions and variations of the basic problem of graph coloring are introduced. The problem consists essentially in finding in a graph G a k-coloring, i.e., a partition V1,…,Vk of the vertex set of G such that, for some specified neighborhood Ñ(v) of each vertex v, the number of vertices in Ñ(v)∩Vi is (at most) a given integer hvi. The complexity of some variations is discussed according to Ñ(v), which may be the usual neighbors, or the vertices at distance at most 2, or the closed neighborhood of v (v and its neighbors). Polynomially solvable cases are exhibited (in particular when G is a special tree)

A discrete model for studying existence and uniqueness of solutions in nonlinear resistive circuits

AbstractTwo combinatorial problems raised by the fundamental question of the existence and uniqueness of solutions in nonlinear electric circuits are presented. The first problem, namely the existence of a pair of conjugate trees, has been solved in polynomial time using an original model based on matroïd intersection. For the second problem, which is the search of a particular orientation in a multigraph with labeled edges, an elaborate branch and bound procedure is proposed