Benjamin Lévêque

Contracting graphs to paths and trees

Vertex deletion and edge deletion problems play a central role in Parameterized Complexity. Examples include classical problems like Feedback Vertex Set, Odd Cycle Transversal, and Chordal Deletion. The study of analogous edge contraction problems has so far been left largely unexplored from a parameterized perspective. We consider two basic problems of this type: Tree Contraction and Path Contraction. These two problems take as input an undirected graph G on n vertices and an integer k, and the task is to determine whether we can obtain an acyclic graph or a path, respectively, by a sequence of at most k edge contractions in G. We present an algorithm with running time 4.98knO(1) for Tree Contraction, based on a variant of the color coding technique of Alon, Yuster and Zwick, and an algorithm with running time 2k+o(k)+nO(1) for Path Contraction. Furthermore, we show that Path Contraction has a kernel with at most 5k+3 vertices, while Tree Contraction does not have a polynomial kernel unless NP ⊆ coNP/poly. We find the latter result surprising, because of the connection between Tree Contraction and Feedback Vertex Set, which is known to have a kernel with 4k2 vertices.

Contracting chordal graphs and bipartite graphs to paths and trees

We study the following two graph modification problems: given a graph G and an integer k, decide whether G can be transformed into a tree or into a path, respectively, using at most k edge contractions. These problems, which we call Tree Contraction and Path Contraction, respectively, are known to be NP-complete in general. We show that on chordal graphs these problems can be solved in O(n+m) and O(nm) time, respectively. As a contrast, both problems remain NP-complete when restricted to bipartite input graphs.

Graphs with maximum degree Δ≥17 and maximum average degree less than 3 are list 2-distance (Δ+2)-colorable

For graphs of bounded maximum average degree, we consider the problem of 2-distance coloring. This is the problem of coloring the vertices while ensuring that two vertices that are adjacent or have a common neighbor receive different colors. It is already known that planar graphs of girth at least 6 and of maximum degree @D are list 2-distance (@D+2)-colorable when @D>=24 (Borodin and Ivanova (2009)) and 2-distance (@D+2)-colorable when @D>=18 (Borodin and Ivanova (2009)). We prove here that @D>=17 suffices in both cases. More generally, we show that graphs with maximum average degree less than 3 and @D>=17 are list 2-distance (@D+2)-colorable. The proof can be transposed to list injective (@D+1)-coloring.

2‐Distance Coloring of Sparse Graphs

A 2-distance coloring of a graph is a coloring of the vertices such that two vertices at distance at most 2 receive distinct colors. We prove that every graph with maximum degree Δ at least 4 and maximum average degree less that 7 admits a 2-distance (Δ + 1)-coloring. This result is tight. This improves previous known results of Dolama and Sopena.

Triangle contact representations and duality

A contact representation by triangles of a graph is a set of triangles in the plane such that two triangles intersect on at most one point, each triangle represents a vertex of the graph and two triangles intersects if and only if their corresponding vertices are adjacent. de Fraysseix, Ossona de Mendez and Rosenstiehl proved that every planar graph admits a contact representation by triangles. We strengthen this in terms of a simultaneous contact representation by triangles of a planar map and of its dual. A primal-dual contact representation by triangles of a planar map is a contact representation by triangles of the primal and a contact representation by triangles of the dual such that for every edge uv, bordering faces f and g, the intersection between the triangles corresponding to u and v is the same point as the intersection between the triangles corresponding to f and g. We prove that every 3-connected planar map admits a primal-dual contact representation by triangles. Moreover, the interiors of the triangles form a tiling of the triangle corresponding to the outer face and each contact point is a node of exactly three triangles. Then we show that these representations are in one-to-one correspondence with generalized Schnyder woods defined by Felsner for 3-connected planar maps.

Toroidal Maps: Schnyder Woods, Orthogonal Surfaces and Straight-Line Representations

A Schnyder wood is an orientation and coloring of the edges of a planar map satisfying a simple local property. We propose a generalization of Schnyder woods to graphs embedded on the torus with application to graph drawing. We prove several properties on this new object. Among all we prove that a graph embedded on the torus admits such a Schnyder wood if and only if it is an essentially 3-connected toroidal map. We show that these Schnyder woods can be used to embed the universal cover of an essentially 3-connected toroidal map on an infinite and periodic orthogonal surface. Finally we use this embedding to obtain a straight-line flat torus representation of any toroidal map in a polynomial size grid.

Contracting Chordal Graphs and Bipartite Graphs to Paths and Trees

Some of the most well studied problems in algorithmic graph theory deal with modifying a graph into an acyclic graph or into a path, using as few operations as possible. In Feedback Vertex Set and Longest Induced Path, the only allowed operation is vertex deletion, and in Spanning Tree and Longest Path, only edge deletions are permitted. We study the edge contraction variant of these problems: given a graph G and an integer k, decide whether G can be transformed into an acyclic graph or into a path using at most k edge contractions. Both problems are known to be NP-complete in general. We show that on chordal graphs these problems can be solved in O(n+m) and O(nm) time, respectively. On the negative side, both problems remain NP-complete when restricted to bipartite input graphs.

2-distance coloring of sparse graphs

Abstract A 2-distance coloring of a graph is a coloring of the vertices such that two vertices at distance at most 2 receive distinct colors. We prove that every graph with maximum degree Δ at least 4 and maximum average degree less that 7 3 admits a 2-distance ( Δ + 1 ) -coloring. This result is tight. This improves previous known results of Dolama and Sopena.

Coloring Bull-Free Perfectly Contractile Graphs

We consider the class of graphs that contain no bull, no odd hole, and no antihole of length at least five. We present a new algorithm that colors optimally the vertices of every graph in this class. This algorithm is based on the existence in every such graph of an ordering of the vertices with a special property. More generally we prove, using a variant of lexicographic breadth-first search, that in every graph that contains no bull and no hole of length at least five there is a vertex that is not the middle of a chordless path on five vertices. This latter fact also generalizes known results about chordal bipartite graphs, totally balanced matrices, and strongly chordal graphs.

List coloring the square of sparse graphs with large degree

We consider the problem of coloring the squares of graphs of bounded maximum average degree, that is, the problem of coloring the vertices while ensuring that two vertices that are adjacent or have a common neighbor receive different colors. Borodin et al. proved in 2004 and 2008 that the squares of planar graphs of girth at least seven and sufficiently large maximum degree @D are list (@D+1)-colorable, while the squares of some planar graphs of girth six and arbitrarily large maximum degree are not. By Eulers Formula, planar graphs of girth at least 6 are of maximum average degree less than 3, and planar graphs of girth at least 7 are of maximum average degree less than 145 =f(m) is list (@D+1)-colorable. Note the planarity assumption is dropped. This bound of 3 is optimal in the sense that the above-mentioned planar graphs with girth 6 have maximum average degree less than 3 and arbitrarily large maximum degree, while their square cannot be (@D+1)-colored. The same holds for list injective @D-coloring.