Publications (249)160.78 Total impact
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ABSTRACT: A graph is $\ell$choosable if, for any choice of lists of $\ell$ colors for each vertex, there is a list coloring, which is a coloring where each vertex receives a color from its list. We study complexity issues of choosability of graphs when the number $k$ of colors is limited. We get results which differ surprisingly from the usual case where $k$ is implicit and which extend known results for the usual case. We also exhibit some classes of graphs (defined by structural properties of their blocks) which are choosable. Finally we show that for any $\ell\geq 3$ and any $k\geq 2\ell2$ there is a bipartite graph which is $\ell$choosable with $k$ colors but not with $k+1$.  [Show abstract] [Hide abstract]
ABSTRACT: Given a positive integer n we find a graph G=(V,E) on V=n vertices with a minimum number of edges such that for any pair of non adjacent vertices x,y the graph Gxy contains a (almost) perfect matching M. Intuitively the non edge xy and M form a (almost) perfect matching of G. Similarly we determine a graph G=(V,E) with a minimum number of edges such that for any matching M¯ of the complement G¯ of G with size ⌊n2⌋1, GV(M¯) contains an edge e. Here M¯ and the edge e of G form a (almost) perfect matching of G¯.We characterize these minimal graphs for all values of n.  [Show abstract] [Hide abstract]
ABSTRACT: An induced matching $M$ in a graph $G$ is dominating if every edge not in $M$ shares exactly one vertex with an edge in $M$. The dominating induced matching problem (also known as efficient edge domination) asks whether a graph $G$ contains a dominating induced matching. This problem is generally NPcomplete, but polynomialtime solvable for graphs with some special properties. In particular, it is solvable in polynomial time for clawfree graphs. In the present paper, we study this problem for graphs containing no long claw, i.e. no induced subgraph obtained from the claw by subdividing each of its edges exactly once. To solve the problem in this class, we reduce it to the following question: given a graph $G$ and a subset of its vertices, does $G$ contain a matching saturating all vertices of the subset? We show that this question can be answered in polynomial time, thus providing a polynomialtime algorithm to solve the dominating induced matching problem for graphs containing no long claw. 
Chapter: Chromatic scheduling
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ABSTRACT: Variations and extensions of the basic vertexcolouring and edgecolouring models have been developed to deal with increasingly complex scheduling problems. We present and illustrate them in specific situations where additional requirements are imposed. We include listcolouring, mixed graph colouring, cocolouring, colouring with preferences and bandwidth colouring, and we present applications of edgecolourings to open shop, school timetabling and sports scheduling problems. We also discuss balancing and compactness constraints which often appear in practical situations. Introduction We show here how graph colouring models may provide a natural tool for dealing with a variety of scheduling problems. Starting from the basic vertexcolouring model, we will introduce some variations and extensions that are motivated by their applications to some scheduling issues. In each case we give references for further results and for extensions of the various models presented. For algorithms, see Chapter 13. In chromatic scheduling problems we have a collection V of items, such as operations of jobs to be performed. In V there are some pairs v, w that are subject to an incompatibility condition and we call E the set of such incompatibility pairs. These data are represented by the graph G = (V, E) in which the items are associated with the vertices and the incompatible pairs v, w with the edges vw between the corresponding vertices. We also have a set C = {1, 2, …, k} of time periods (of unit duration). Assuming that each item (considered as an operation) has unit completion time, we may ask whether we can find a schedule taking the incompatibilities into account and using at most k periods of time. This is precisely the vertex kcolouring problem: there exists a feasible schedule if and only if the set V of vertices can be partitioned into subsets S1, S2, …, Sk, where each Si contains no two incompatible items. In some instances, we may try to find the smallest set C of periods (that is, the smallest k) for which a schedule in time k = C exists.  [Show abstract] [Hide abstract]
ABSTRACT: The notion of augmenting graphs generalizes Berge's idea of augmenting chains, which was used by Edmonds in his celebrated solution of the maximum matching problem. This problem is a special case of the more general maximum independent set (MIS) problem. Recently, the augmenting graph approach has been successfully applied to solve MIS in various other special cases. However, our knowledge of augmenting graphs is still very limited, and we do not even know what the minimal infinite classes of augmenting graphs are. In the present paper, we find an answer to this question and apply it to extend the area of polynomialtime solvability of the maximum independent set problem. 
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ABSTRACT: We define as extensible a graph G such that for every pair u,v of non adjacent vertices it is possible to extend the nonedge uv to a perfect (or near perfect) matching using only edges of G that are not incident to u or v. For every order n of G we give Ext(n) the minimum size of an extensible graph.  [Show abstract] [Hide abstract]
ABSTRACT: A threedimensional Nuclear Magnetic Resonance (NMR) map displays the results of NMR experiments, that allow to determine the shape of a biological molecule. Shape calculation starts from a reconstruction of a sequence of NMR signals, which is equivalent to finding a specific path in a graph representation of the problem. Let G=(V,E)G=(V,E) be a graph that models the interactions reflected on an NMR map. Its edges are colored with cc colors, where each color corresponds to one of cc different relationships between the signals. The sequence of interactions under consideration is represented using a concept of an orderly colored path in the ccedgecolored graph. In this paper, we consider the problem of finding the required arrangement of NMR signals on the 3D map and we present its graph representation. We discuss the computational complexity of the problem, we consider its two alternative integer programming models, and evaluate the performance of an optimization algorithm based on the solution of their relaxation combined with the separation of fractional cycles in a Branch & Cut scheme. 
Article: Combinatorial Optimization Preface


Article: Graph Coloring Problems
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ABSTRACT: We present the basic concepts of colorings as well as a series of variations and generalizations prompted by various scheduling problems including drawing up school timetables. A few recent exact algorithms and some heuristics will be presented. In particular we will give an outline of methods based on the tabu search for finding approximate solutions for large problems. Lastly, we mention application of colorings to various problems, including computer file transfers and production systems. This text is an extended version of [D. de Werra and D. Kobler, RAIRO, Oper. Res. 37, No. 1, 29–66 (2003; Zbl 1062.90026)]. 
Article: On some coloring problems in grids
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ABSTRACT: We study complexity issues related to some coloring problems in grids: we examine in particular the case of List coloring, of Precoloring extension and of (p,q)(p,q)List coloring, the case of List coloring in bipartite graphs where lists in the first part of the bipartition are all of size pp and lists in the second part are of size qq. In particular, we characterize the complexity of (p,q)(p,q)List coloring in grid graphs, showing that the only NPcomplete case is (2, 3)List coloring with k≥4k≥4 colors. We also show that Precoloring extension with 3 colors is NPcomplete in subgrids.  [Show abstract] [Hide abstract]
ABSTRACT: Let G=(V,E) be a graph in which every vertex v∈V has a weight w(v)⩾0 and a cost c(v)⩾0. Let SG be the family of all maximumweight stable sets in G. For any integer d⩾0, a minimum dtransversal in the graph G with respect to SG is a subset of vertices T⊆V of minimum total cost such that T∩S⩾d for every S∈SG. In this paper, we present a polynomialtime algorithm to determine minimum dtransversals in bipartite graphs. Our algorithm is based on a characterization of maximumweight stable sets in bipartite graphs. We also derive results on minimum dtransversals of minimumweight vertex covers in weighted bipartite graphs.  [Show abstract] [Hide abstract]
ABSTRACT: In threshold graphs one may find weights for the vertices and a threshold value t such that for any subset S of vertices, the sum of the weights is at most the threshold t if and only if the set S is a stable (independent) set. In this note we ask a similar question about vertex colorings: given an integer p, when is it possible to find weights (in general depending on p) for the vertices and a threshold value tp such that for any subset S of vertices the sum of the weights is at most tp if and only if S generates a subgraph with chromatic number at most p−1? We show that threshold graphs do have this property and we show that one can even find weights which are valid for all values of p simultaneously. 
Conference Paper: (p,q)  Choosability of Grid Graphs
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ABSTRACT: A general formulation of the problems we are going to consider may be sketched as follows: we are given a system S which is operated by an actor A; this actor tries to choose among several optimal actions which may be represented by subsets of S. An opponent O wants to prevent actor A from operating S in an optimum way by destroying some part P of S. O may in particular wish to find a part P of S as small as possible whose removal will reduce the efficiency of the operation of the system S by a given amount. Another way for O would be to determine a smallest possible part P (the most vital elements) which hits in a sufficient amount every possible optimal action of A. This kind of problem occurs in various situation related to safety or reliability or even in game theoretic contexts. Such problems have been studied from a theoretical point of view in very special cases for which combinatorial optimization models may give an acceptable represen tation of S. It leads to challenging optimization problems; the goal of this chapter is to give a partial survey of such situations while focusing on simple models based on graphs and other (hopefully tractable) combinatorial structures.  [Show abstract] [Hide abstract]
ABSTRACT: We consider a set V of elements and an optimization problem on V: the search for a maximum (or minimum) cardinality subset of V verifying a given property ℘. A dtransversal is a subset of V which intersects any optimum solution in at least d elements while a dblocker is a subset of V whose removal deteriorates the value of an optimum solution by at least d. We present some general characteristics of these problems, we review some situations which have been studied (matchings, s–t paths and s–t cuts in graphs) and we study dtransversals and dblockers of stable sets or vertex covers in bipartite and in split graphs. 
Conference Paper: Minimum dTransversals of MaximumWeight Stable Sets in Trees
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ABSTRACT: Given an integer d and a weighted tree T, we show how to find in polynomial time a minimum dtransversal of all maximumweight stable sets in T, i.e., a set of vertices of minimum size having at least d vertices in common with every maximumweight stable set. Our proof relies on new structural results for maximumweight stable sets on trees. 

Publication Stats
4k  Citations  
160.78  Total Impact Points  
Top Journals
Institutions

19732014

École Polytechnique Fédérale de Lausanne
 School of Basic Sciences
Lausanne, Vaud, Switzerland


2002

École Polytechnique
Paliseau, ÎledeFrance, France


19841996

Eawag: Das WasserforschungsInstitut des ETHBereichs
Duebendorf, Zurich, Switzerland 
Poznan University of Technology
Posen, Greater Poland Voivodeship, Poland


1988

Beijing Institute Of Technology
Peping, Beijing, China


1974

Ecole polytechnique fédérale de Lausanne
Lausanne, Vaud, Switzerland
