Université de Fribourg

Bicolored Matchings in Some Classes of Graphs

Costa, Marie-Christine ; de Werra, Dominique ; Picouleau, Christophe ; Ries, Bernard

In: Graphs and Combinatorics, 2007, vol. 23, no. 1, p. 47-60

We consider the problem of finding in a graph a set R of edges to be colored in red so that there are maximum matchings having some prescribed numbers of red edges. For regular bipartite graphs with n nodes on each side, we give sufficient conditions for the existence of a set R with |R| = n + 1 such that perfect matchings with k red edges exist for all k, 0 ≤ k ≤ n. Given two integers p...

Université de Fribourg

On a graph coloring problem arising from discrete tomography

Bentz, Cédric ; Costa, Marie-Christine ; de Werra, Dominique ; Picouleau, Christophe ; Ries, Bernard

In: Networks, 2008, vol. 51, no. 4, p. 256-267

An extension of the basic image reconstruction problem in discrete tomography is considered: given a graph G = (V,E) and a family equation image of chains Pi together with vectors h(Pi) = (h1, . . . , hik), one wants to find a partition V1,…,Vk of V such that for each Pi and each color j, |Vj ∩ Pi| = hij. An interpretation in terms of scheduling is presented. We consider special cases of...

Université de Fribourg

Degree-constrained edge partitioning in graphs arising from discrete tomography

Bentz, Cédric ; Costa, Marie-Christine ; Picouleau, Christophe ; Ries, Bernard ; de Werra, Dominique

In: Journal of Graph Algorithms and Applications, 2009, vol. 13, no. 2, p. 99-118

Starting from the basic problem of reconstructing a 2-dimensional im- age given by its projections on two axes, one associates a model of edge coloring in a complete bipartite graph. The complexity of the case with k = 3 colors is open. Variations and special cases are considered for the case k = 3 colors where the graph corresponding to the union of some color classes (for instance colors 1...

Université de Fribourg

Blockers and transversals in some subclasses of bipartite graphs : When caterpillars are dancing on a grid

Ries, Bernard ; Bentz, Cédric ; Picouleau, Christophe ; de Werra, Dominique ; Costa, Marie-Christine ; Zenklusen, Rico

In: Discrete Mathematics, 2010, vol. 310, p. 132-146

Given an undirected graph G=(V,E) with matching number \nu(G), a d-blocker is a subset of edges B such that \nu(/V,E\B))= d. While the associated decision problem is NP-complete in bipartite graphs we show how to construct efficiently minimum d-transversals and minimum d-blockers in the...

Université de Fribourg

d-Transversals of stable sets and vertex covers in weighted bipartite graphs

Bentz, Cédric ; Costa, Marie-Christine ; Picouleau, Christophe ; Ries, Bernard ; de Werra, Dominique

In: Journal of Discrete Algorithms, 2012, vol. 17, p. 95-102

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 maximum-weight stable sets in G. For any integer d 0, a minimum d-transversal 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 polynomial-time algorithm to...

Université de Fribourg

Blockers and transversals

Zenklusen, Rico ; Ries, Bernard ; Picouleau, Christophe ; de Werra, Dominique ; Costa, Marie-Christine ; Bentz, Cédric

In: Discrete Mathematics, 2009, vol. 309, p. 4306-4314

Given an undirected graph G=(V,E) with matching number \nu(G), we define d- blockers as subsets of edges B such that \nu(G=(V,E\B))\leq \nu(G)-d. We define d- transversals T as subsets of edges such that every maximum matching M has |M\cap T|\geq d. We explore connections between d-blockers and d-transversals. Special classes of graphs are examined which include complete graphs, regular...

Université de Fribourg

Graph coloring with cardinality constraints on the neighborhoods

Costa, Marie-Christine ; de Werra, Dominique ; Picouleau, Christophe ; Ries, Bernard

In: discrete Optimization, 2009, vol. 6, no. 4, p. 362-369

Extensions and variations of the basic problem of graph coloring are introduced. The problem consists essentially in finding in a graph a k-coloring, i.e., a partition (V_1,\cdots,V_k) of the vertex set of G such that, for some specified neighborhood \tilde|{N}(v) of each vertex v, the number of vertices in \tilde|{N}(v)\cap V_i is (at most) a given integer h_i^v. The complexity of some...

Université de Fribourg

Reducing the Clique and Chromatic Number via Edge Contractions and Vertex Deletions

Paulusma, Daniël ; Picouleau, Christophe ; Ries, Bernard

In: Lecture Notes in Computer Science, 2016, vol. 9849, p. 38-49

We consider the following problem: can a certain graph parameter of some given graph G be reduced by at least d, for some integer d, via at most k graph operations from some specified set S, for some given integer k? As graph parameters we take the chromatic number and the clique number. We let the set S consist of either an edge contraction or a vertex deletion. As all these problems are...

Université de Fribourg

Blocking Independent Sets for H-Free Graphs via Edge Contractions and Vertex Deletions

Paulusma, Daniël ; Picouleau, Christophe ; Ries, Bernard

In: Lecture Notes in Computer Science, 2017, vol. 10185, p. 470-483

Let d and k be two given integers, and let G be a graph. Can we reduce the independence number of G by at least d via at most k graph operations from some fixed set S? This problem belongs to a class of so-called blocker problems. It is known to be co-NP-hard even if S consists of either an edge contraction or a vertex deletion. We further investigate its computational complexity under these...

Public access from Oct 11, 2020
Université de Fribourg

Critical vertices and edges in H-free graphs

Paulusma, Daniël ; Picouleau, Christophe ; Ries, Bernard

In: Discrete applied mathematics, 2019, vol. 257, p. 361-367

A vertex or edge in a graph is critical if its deletion reduces the chromatic number of the graph by one. We consider the problems of deciding whether a graph has a critical vertex or edge, respectively. We give a complexity dichotomy for both problems restricted to H-free graphs, that is, graphs with no induced subgraph isomorphic to H. Moreover, we show that an edge is critical if and only...