Mathieu Liedloff

On Independent Sets and Bicliques in Graphs

Bicliques of graphs have been studied extensively, partially motivated by the large number of applications. In this paper we improve Prisner’s upper bound on the number of maximal bicliques (Combinatorica, 20, 109–117, 2000) and show that the maximum number of maximal bicliques in a graph on n vertices is Θ(3n/3). Our major contribution is an exact exponential-time algorithm. This branching algorithm computes the number of distinct maximal independent sets in a graph in time O(1.3642n), where n is the number of vertices of the input graph. We use this counting algorithm and previously known algorithms for various other problems related to independent sets to derive algorithms for problems related to bicliques via polynomial-time reductions.

An exact algorithm for the Maximum Leaf Spanning Tree problem

Given an undirected graph with n vertices, the Maximum Leaf Spanning Tree problem is to find a spanning tree with as many leaves as possible. When parameterized in the number of leaves k, this problem can be solved in time O(4kpoly(n)) using a simple branching algorithm introduced by a subset of the authors (Kneis et al. 2008 16). Daligault et al. (2010) 6 improved the branching and obtained a running time of O(3.72kpoly(n)). In this paper, we study the problem from an exponential time viewpoint, where it is equivalent to the Connected Dominating Set problem. Here, Fomin, Grandoni, and Kratsch showed how to break the ?(2n) barrier and proposed an O(1.9407n)-time algorithm (Fomin et al. 2008 11). Based on some useful properties of Kneis et al. (2008) 16 and Daligault et al. (2010) 6, we present a branching algorithm whose running time of O(1.8966n) has been analyzed using the Measure-and-Conquer technique. Finally, we provide a lower bound of ?(1.4422n) for the worst case running time of our algorithm.

Exact and Parameterized Algorithms for Max Internal Spanning Tree

We consider the

Exact Algorithms for L (2,1)-Labeling of Graphs

\mathcal{NP}

Iterative Compression and Exact Algorithms

-hard problem of finding a spanning tree with a maximum number of internal vertices. This problem is a generalization of the famous Hamiltonian Path problem. Our dynamic-programming algorithms for general and degree-bounded graphs have running times of the form

Breaking the 2n-barrier for Irredundance: Two lines of attack

\mathcal{O}^{*}(c^{n})

An Exact Algorithm for the Maximum Leaf Spanning Tree Problem

with c≤2. For graphs with bounded degree, c<2. The main result, however, is a branching algorithm for graphs with maximum degree three. It only needs polynomial space and has a running time of

Exact algorithms for L (2, 1)-labeling of graphs

\mathcal{O}(1.8612^{n})

Roman domination over some graph classes

when analyzed with respect to the number of vertices. We also show that its running time is

On the number of minimal dominating sets on some graph classes

2.1364^{k} n^{\mathcal{O}(1)}