11 research outputs found
Long induced paths in graphs
We prove that every 3-connected planar graph on vertices contains an
induced path on vertices, which is best possible and improves
the best known lower bound by a multiplicative factor of . We
deduce that any planar graph (or more generally, any graph embeddable on a
fixed surface) with a path on vertices, also contains an induced path on
vertices. We conjecture that for any , there is a
contant such that any -degenerate graph with a path on vertices
also contains an induced path on vertices. We provide
examples showing that this order of magnitude would be best possible (already
for chordal graphs), and prove the conjecture in the case of interval graphs.Comment: 20 pages, 5 figures - revised versio
Fine-grained parameterized complexity analysis of graph coloring problems
The q-COLORING problem asks whether the vertices of a graph can be properly colored with q colors. In this paper we perform a fine-grained analysis of the complexity of q-COLORING with respect to a hierarchy of structural parameters. We show that unless the Exponential Time Hypothesis fails, there is no constant θ such that q-COLORING parameterized by the size k of a vertex cover can be solved in O ∗(θ k) time for all fixed q. We prove that there are O ∗((q−ɛ) k) time algorithms where k is the vertex deletion distance to several graph classes for which q-COLORING is known to be solvable in polynomial time, including all graph classes F whose (q+1)-colorable members have bounded treedepth. In contrast, we prove that if F is the class of paths – some of the simplest graphs of unbounded treedepth – then no such algorithm can exist unless the Strong Exponential Time Hypothesis fails.</p
Fine-grained parameterized complexity analysis of graph coloring problems
The q-COLORING problem asks whether the vertices of a graph can be properly colored with q colors. In this paper we perform a fine-grained analysis of the complexity of q-COLORING with respect to a hierarchy of structural parameters. We show that unless the Exponential Time Hypothesis fails, there is no constant θ such that q-COLORING parameterized by the size k of a vertex cover can be solved in O ∗(θ k) time for all fixed q. We prove that there are O ∗((q−ɛ) k) time algorithms where k is the vertex deletion distance to several graph classes for which q-COLORING is known to be solvable in polynomial time, including all graph classes F whose (q+1)-colorable members have bounded treedepth. In contrast, we prove that if F is the class of paths – some of the simplest graphs of unbounded treedepth – then no such algorithm can exist unless the Strong Exponential Time Hypothesis fails.</p
Linear time algorithm for computing a small biclique in graphs without long induced paths
The biclique problem asks, given a graph G and a parameter k, whether G has a complete bipartite subgraph of k vertices in each part (a biclique of order k). Fixed-parameter tractability of this problem is a longstanding open question in parameterized complexity that received a lot of attention from the community. In this paper we consider a restricted version of this problem by introducing an additional parameter s and assuming that G does not have induced (i.e. chordless) paths of length s. We prove that under this parameterization the problem becomes fixed-parameter linear. The main tool in our proof is a Ramsey-type theorem stating that a graph with a long (not necessarily induced) path contains either a long induced path or a large biclique
Hereditary classes of graphs : a parametric approach
The world of hereditary classes is rich and diverse and it contains a variety of classes of theoretical and practical importance. Thousands of results in the literature are devoted to individual classes and only a few of them analyse the universe of hereditary classes as a whole. To shift the analysis into a new level, in the present paper we exploit an approach, where we operate by infinite families of classes, rather than individual classes. Each family is associated with a graph parameter and is characterized by classes that are critical with respect to the parameter. In particular, we obtain a complete parametric description of the bottom of the lattice of hereditary classes and discuss a number of open questions related to this approach