The four-in-a-tree problem in triangle-free graphs

The three-in-a-tree algorithm of Chudnovsky and Seymour decides in time O(n4) whether three given vertices of a graph belong to an induced tree. Here, we study four-in-a-tree for triangle-free graphs. We give a structural answer to the following question : how does look like a triangle-free graph such that no induced tree covers four given vertices ? Our main result says that any such graph must have the "same structure", in a sense to be defined precisely, as a square or a cube. We provide an O(nm)-time algorithm that given a triangle-free graph G together with four vertices outputs either an induced tree that contains them or a partition of V(G) certifying that no such tree exists. We prove that the problem of deciding whether there exists a tree T covering the four vertices such that at most one vertex of T has degree at least 3 is NP-complete.Three, four, tree, algorithm, 3-in-a-tree, 4-in-a-tree, triangle-free graphs.

The four-in-a-tree problem in triangle-free graphs

URL des Documents de travail : http://ces.univ-paris1.fr/cesdp/cesdp2008.htmlDocuments de travail du Centre d'Economie de la Sorbonne 2008.23 - ISSN : 1955-611XThe three-in-a-tree algorithm of Chudnovsky and Seymour decides in time O(n4) whether three given vertices of a graph belong to an induced tree. Here, we study four-in-a-tree for triangle-free graphs. We give a structural answer to the following question : how does look like a triangle-free graph such that no induced tree covers four given vertices ? Our main result says that any such graph must have the "same structure", in a sense to be defined precisely, as a square or a cube. We provide an O(nm)-time algorithm that given a triangle-free graph G together with four vertices outputs either an induced tree that contains them or a partition of V(G) certifying that no such tree exists. We prove that the problem of deciding whether there exists a tree T covering the four vertices such that at most one vertex of T has degree at least 3 is NP-complete.L'algorithme "three-in-a-tree" de Chudnovsky et Seymour décide en temps O(n4) si trois sommets donnés d'un graphe appartiennent à un arbre induit. Ici, nous étudions four-in-a-tree pour les graphes sans triangles. Nous donnons une réponse structurelle à la question suivante : à quoi ressemble un graphe sans triangles dont aucun arbre induit ne couvre quatre sommets donnés ? Notre résultat principal affirme qu'un tel graphe doit avoir la "même structure", dans un sens précisemment défini, qu'un carré ou un cube. Nous donnons un algorithme en temps O(nm) qui étant donné un graphe sans triangles G et quatre de ses sommets, retourne ou bien un arbre induit qui contient les sommets, ou bien une partition de V(G) certifiant qu'un tel arbre n'existe pas. Nous prouvons que le problème consistant à décider s'il existe un arbre T couvrant les quatre sommets et tel qu'au plus un sommet de T est de degré au moins 3 est NP-complet

Planar Induced Subgraphs of Sparse Graphs

We show that every graph has an induced pseudoforest of at least $n-m/4.5$ vertices, an induced partial 2-tree of at least $n-m/5$ vertices, and an induced planar subgraph of at least $n-m/5.2174$ vertices. These results are constructive, implying linear-time algorithms to find the respective induced subgraphs. We also show that the size of the largest $K_h$-minor-free graph in a given graph can sometimes be at most $n-m/6+o(m)$.Comment: Accepted by Graph Drawing 2014. To appear in Journal of Graph Algorithms and Application

Some results on triangle partitions

We show that there exist efficient algorithms for the triangle packing problem in colored permutation graphs, complete multipartite graphs, distance-hereditary graphs, k-modular permutation graphs and complements of k-partite graphs (when k is fixed). We show that there is an efficient algorithm for C_4-packing on bipartite permutation graphs and we show that C_4-packing on bipartite graphs is NP-complete. We characterize the cobipartite graphs that have a triangle partition

Planar graph coloring avoiding monochromatic subgraphs: trees and paths make things difficult

We consider the problem of coloring a planar graph with the minimum number of colors such that each color class avoids one or more forbidden graphs as subgraphs. We perform a detailed study of the computational complexity of this problem

Coloring triangle-free rectangle overlap graphs with O(log⁡log⁡n)O(\log\log n) colors

Recently, it was proved that triangle-free intersection graphs of $n$ line segments in the plane can have chromatic number as large as $\Theta(\log\log n)$. Essentially the same construction produces $\Theta(\log\log n)$-chromatic triangle-free intersection graphs of a variety of other geometric shapes---those belonging to any class of compact arc-connected sets in $\mathbb{R}^2$ closed under horizontal scaling, vertical scaling, and translation, except for axis-parallel rectangles. We show that this construction is asymptotically optimal for intersection graphs of boundaries of axis-parallel rectangles, which can be alternatively described as overlap graphs of axis-parallel rectangles. That is, we prove that triangle-free rectangle overlap graphs have chromatic number $O(\log\log n)$, improving on the previous bound of $O(\log n)$. To this end, we exploit a relationship between off-line coloring of rectangle overlap graphs and on-line coloring of interval overlap graphs. Our coloring method decomposes the graph into a bounded number of subgraphs with a tree-like structure that "encodes" strategies of the adversary in the on-line coloring problem. Then, these subgraphs are colored with $O(\log\log n)$ colors using a combination of techniques from on-line algorithms (first-fit) and data structure design (heavy-light decomposition).Comment: Minor revisio