Linear Transformations Between Dominating Sets in the TAR-Model

Given a graph $G$ and an integer $k$, a token addition and removal ({\sf TAR} for short) reconfiguration sequence between two dominating sets $D_{\sf s}$ and $D_{\sf t}$ of size at most $k$ is a sequence $S= \langle D_0 = D_{\sf s}, D_1 \ldots, D_\ell = D_{\sf t} \rangle$ of dominating sets of $G$ such that any two consecutive dominating sets differ by the addition or deletion of one vertex, and no dominating set has size bigger than $k$. We first improve a result of Haas and Seyffarth, by showing that if $k=\Gamma(G)+\alpha(G)-1$ (where $\Gamma(G)$ is the maximum size of a minimal dominating set and $\alpha(G)$ the maximum size of an independent set), then there exists a linear {\sf TAR} reconfiguration sequence between any pair of dominating sets. We then improve these results on several graph classes by showing that the same holds for $K_{\ell}$-minor free graph as long as $k \ge \Gamma(G)+O(\ell \sqrt{\log \ell})$ and for planar graphs whenever $k \ge \Gamma(G)+3$. Finally, we show that if $k=\Gamma(G)+tw(G)+1$, then there also exists a linear transformation between any pair of dominating sets.Comment: 13 pages, 6 figure

Reconfiguration of Spanning Trees with Degree Constraint or Diameter Constraint

We investigate the complexity of finding a transformation from a given spanning tree in a graph to another given spanning tree in the same graph via a sequence of edge flips. The exchange property of the matroid bases immediately yields that such a transformation always exists if we have no constraints on spanning trees. In this paper, we wish to find a transformation which passes through only spanning trees satisfying some constraint. Our focus is bounding either the maximum degree or the diameter of spanning trees, and we give the following results. The problem with a lower bound on maximum degree is solvable in polynomial time, while the problem with an upper bound on maximum degree is PSPACE-complete. The problem with a lower bound on diameter is NP-hard, while the problem with an upper bound on diameter is solvable in polynomial time

Reconfiguration of Spanning Trees with Many or Few Leaves

Let G be a graph and T?,T? be two spanning trees of G. We say that T? can be transformed into T? via an edge flip if there exist two edges e ? T? and f in T? such that T? = (T??e) ? f. Since spanning trees form a matroid, one can indeed transform a spanning tree into any other via a sequence of edge flips, as observed in [Takehiro Ito et al., 2011]. We investigate the problem of determining, given two spanning trees T?,T? with an additional property ?, if there exists an edge flip transformation from T? to T? keeping property ? all along. First we show that determining if there exists a transformation from T? to T? such that all the trees of the sequence have at most k (for any fixed k ? 3) leaves is PSPACE-complete. We then prove that determining if there exists a transformation from T? to T? such that all the trees of the sequence have at least k leaves (where k is part of the input) is PSPACE-complete even restricted to split, bipartite or planar graphs. We complete this result by showing that the problem becomes polynomial for cographs, interval graphs and when k = n-2

Reconfiguration problems in graphs

Dans cette thèse, nous nous intéressons à la théorie des graphes, et plus particulièrement à des problèmes de reconfiguration. Pour un problème d'optimisation donné, l'objectif est alors d'étudier les relations existant entre les différentes solutions. Typiquement, est-il possible de transformer étape par étape une solution en une autre à l'aide d'opérations élémentaires, de telle sorte que chaque étape intermédiaire soit également une solution ?Le problème au départ de cette thèse est celui d'Ensemble Dominant, qui consiste à trouver un sous-ensemble D de sommets tel que chaque sommet est dans D ou adjacent à un sommet de D. Nous étudions la reconfiguration d'ensembles dominants sous deux opérations élémentaires différentes, principalement d'un point de vue algorithmique. Nous donnons également des conditions nécessaires et suffisantes garantissant qu'une transformation est toujours possible entre deux solutions données. Enfin, nous nous intéressons à la complexité paramétrée d'une variante d'optimisation : étant donné un ensemble dominant D, quel est le plus petit ensemble dominant que l'on peut atteindre depuis D sous certaines contraintes ? Nous nous intéressons également à deux autres questions de reconfiguration. Nous étudions d'une part la complexité de la reconfiguration d'arbres couvrant avec une contrainte sur le nombre minimum de feuilles ; d'autre part la recoloration dans le modèle LOCAL, un modèle de calcul distribué. Pour cette dernière question, nous cherchons à optimiser à la fois le nombre de communications et d'étapes permettant de transformer une coloration en une autre.In this thesis, we are interested in graph theory, and more specifically in reconfiguration problems. The goal of this area is to study the relationship between the feasible solutions of a given combinatorial optimization problem.Typically, is it possible to find a step-by-step transformation between two solutions thanks to an elementary operation?The original problem of this thesis is the so-called Dominating Set problem, which consists in finding a subset D of vertices such that each vertex either belongs to D or is adjacent to a vertex in D. We study the reconfiguration of dominating sets under two different elementary operations, mainly from an algorithmic point of view. We also provide necessary and sufficient conditions to ensure that a transformation always exists between two given solutions. Finally, we are interested in the parameterized complexity of an optimization variant: given a dominating set D, what is the smallest dominating set that is reachable from D under certain constraints?We are also interested in two other reconfiguration problems. First, we study the complexity of spanning trees reconfiguration with some constraints with respect to the minimum number of leaves. Finally, we introduce recoloring in the LOCAL model in Distributed Computing. In this last problem, we seek to optimize both the number of communication rounds and the number of steps between the two colorings

Problèmes de reconfiguration dans les graphes

In this thesis, we are interested in graph theory, and more specifically in reconfiguration problems. The goal of this area is to study the relationship between the feasible solutions of a given combinatorial optimization problem.Typically, is it possible to find a step-by-step transformation between two solutions thanks to an elementary operation?The original problem of this thesis is the so-called Dominating Set problem, which consists in finding a subset D of vertices such that each vertex either belongs to D or is adjacent to a vertex in D. We study the reconfiguration of dominating sets under two different elementary operations, mainly from an algorithmic point of view. We also provide necessary and sufficient conditions to ensure that a transformation always exists between two given solutions. Finally, we are interested in the parameterized complexity of an optimization variant: given a dominating set D, what is the smallest dominating set that is reachable from D under certain constraints?We are also interested in two other reconfiguration problems. First, we study the complexity of spanning trees reconfiguration with some constraints with respect to the minimum number of leaves. Finally, we introduce recoloring in the LOCAL model in Distributed Computing. In this last problem, we seek to optimize both the number of communication rounds and the number of steps between the two colorings.Dans cette thèse, nous nous intéressons à la théorie des graphes, et plus particulièrement à des problèmes de reconfiguration. Pour un problème d'optimisation donné, l'objectif est alors d'étudier les relations existant entre les différentes solutions. Typiquement, est-il possible de transformer étape par étape une solution en une autre à l'aide d'opérations élémentaires, de telle sorte que chaque étape intermédiaire soit également une solution ?Le problème au départ de cette thèse est celui d'Ensemble Dominant, qui consiste à trouver un sous-ensemble D de sommets tel que chaque sommet est dans D ou adjacent à un sommet de D. Nous étudions la reconfiguration d'ensembles dominants sous deux opérations élémentaires différentes, principalement d'un point de vue algorithmique. Nous donnons également des conditions nécessaires et suffisantes garantissant qu'une transformation est toujours possible entre deux solutions données. Enfin, nous nous intéressons à la complexité paramétrée d'une variante d'optimisation : étant donné un ensemble dominant D, quel est le plus petit ensemble dominant que l'on peut atteindre depuis D sous certaines contraintes ? Nous nous intéressons également à deux autres questions de reconfiguration. Nous étudions d'une part la complexité de la reconfiguration d'arbres couvrant avec une contrainte sur le nombre minimum de feuilles ; d'autre part la recoloration dans le modèle LOCAL, un modèle de calcul distribué. Pour cette dernière question, nous cherchons à optimiser à la fois le nombre de communications et d'étapes permettant de transformer une coloration en une autre

Dominating sets reconfiguration under token sliding

Let $G$ be a graph and $D_{\sf s}$ and $D_{\sf t}$ be two dominating sets of $G$ of size $k$. Does there exist a sequence $\langle D_0 = D_{\sf s}, D_1, \ldots, D_{\ell-1}, D_\ell = D_{\sf t} \rangle$ of dominating sets of $G$ such that $D_{i+1}$ can be obtained from $D_i$ by replacing one vertex with one of its neighbors? In this paper, we investigate the complexity of this decision problem. We first prove that this problem is PSPACE-complete, even when restricted to split, bipartite or bounded treewidth graphs. On the other hand, we prove that it can be solved in polynomial time on dually chordal graphs (a superclass of both trees and interval graphs) or cographs