Hitting forbidden minors: Approximation and Kernelization

We study a general class of problems called F-deletion problems. In an F-deletion problem, we are asked whether a subset of at most $k$ vertices can be deleted from a graph $G$ such that the resulting graph does not contain as a minor any graph from the family F of forbidden minors. We obtain a number of algorithmic results on the F-deletion problem when F contains a planar graph. We give (1) a linear vertex kernel on graphs excluding $t$-claw $K_{1,t}$, the star with $t$ leves, as an induced subgraph, where $t$ is a fixed integer. (2) an approximation algorithm achieving an approximation ratio of $O(\log^{3/2} OPT)$, where $OPT$ is the size of an optimal solution on general undirected graphs. Finally, we obtain polynomial kernels for the case when F contains graph $\theta_c$ as a minor for a fixed integer $c$. The graph $\theta_c$ consists of two vertices connected by $c$ parallel edges. Even though this may appear to be a very restricted class of problems it already encompasses well-studied problems such as {\sc Vertex Cover}, {\sc Feedback Vertex Set} and Diamond Hitting Set. The generic kernelization algorithm is based on a non-trivial application of protrusion techniques, previously used only for problems on topological graph classes

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

Embedding large subgraphs into dense graphs

What conditions ensure that a graph G contains some given spanning subgraph H? The most famous examples of results of this kind are probably Dirac's theorem on Hamilton cycles and Tutte's theorem on perfect matchings. Perfect matchings are generalized by perfect F-packings, where instead of covering all the vertices of G by disjoint edges, we want to cover G by disjoint copies of a (small) graph F. It is unlikely that there is a characterization of all graphs G which contain a perfect F-packing, so as in the case of Dirac's theorem it makes sense to study conditions on the minimum degree of G which guarantee a perfect F-packing. The Regularity lemma of Szemeredi and the Blow-up lemma of Komlos, Sarkozy and Szemeredi have proved to be powerful tools in attacking such problems and quite recently, several long-standing problems and conjectures in the area have been solved using these. In this survey, we give an outline of recent progress (with our main emphasis on F-packings, Hamiltonicity problems and tree embeddings) and describe some of the methods involved

Kernelization and Parameterized Algorithms for 3-Path Vertex Cover

A 3-path vertex cover in a graph is a vertex subset $C$ such that every path of three vertices contains at least one vertex from $C$. The parameterized 3-path vertex cover problem asks whether a graph has a 3-path vertex cover of size at most $k$. In this paper, we give a kernel of $5k$ vertices and an $O^*(1.7485^k)$-time and polynomial-space algorithm for this problem, both new results improve previous known bounds.Comment: in TAMC 2016, LNCS 9796, 201

Partitioning Perfect Graphs into Stars

The partition of graphs into "nice" subgraphs is a central algorithmic problem with strong ties to matching theory. We study the partitioning of undirected graphs into same-size stars, a problem known to be NP-complete even for the case of stars on three vertices. We perform a thorough computational complexity study of the problem on subclasses of perfect graphs and identify several polynomial-time solvable cases, for example, on interval graphs and bipartite permutation graphs, and also NP-complete cases, for example, on grid graphs and chordal graphs.Comment: Manuscript accepted to Journal of Graph Theor

A transfer principle and applications to eigenvalue estimates for graphs

In this paper, we prove a variant of the Burger-Brooks transfer principle which, combined with recent eigenvalue bounds for surfaces, allows to obtain upper bounds on the eigenvalues of graphs as a function of their genus. More precisely, we show the existence of a universal constants $C$ such that the $k$-th eigenvalue $\lambda_k^{nr}$ of the normalized Laplacian of a graph $G$ of (geometric) genus $g$ on $n$ vertices satisfies $$\lambda_k^{nr}(G) \leq C \frac{d_{\max}(g+k)}{n},$$ where $d_{\max}$ denotes the maximum valence of vertices of the graph. This result is tight up to a change in the value of the constant $C$, and improves recent results of Kelner, Lee, Price and Teng on bounded genus graphs. To show that the transfer theorem might be of independent interest, we relate eigenvalues of the Laplacian on a metric graph to the eigenvalues of its simple graph models, and discuss an application to the mesh partitioning problem, extending pioneering results of Miller-Teng-Thurston-Vavasis and Spielman-Tang to arbitrary meshes.Comment: Major revision, 16 page