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

On the independent domination number of graphs with given minimum degree

AbstractWe prove a new upper bound on the independent domination number of graphs in terms of the number of vertices and the minimum degree. This bound is slightly better than that of Haviland (1991) and settles the case δ = 2 of the corresponding conjecture by Favaron (1988)

Dense graphs have K3,t minors

AbstractLet K3,t∗ denote the graph obtained from K3,t by adding all edges between the three vertices of degree t in it. We prove that for each t≥6300 and n≥t+3, each n-vertex graph G with e(G)>12(t+3)(n−2)+1 has a K3,t∗-minor. The bound is sharp in the sense that for every t, there are infinitely many graphs G with e(G)=12(t+3)(|V(G)|−2)+1 that have no K3,t-minor. The result confirms a partial case of the conjecture by Woodall and Seymour that every (s+t)-chromatic graph has a Ks,t-minor

Faster Separators for Shallow Minor-Free Graphs via Dynamic Approximate Distance Oracles

Plotkin, Rao, and Smith (SODA'97) showed that any graph with $m$ edges and $n$ vertices that excludes $K_h$ as a depth $O(\ell\log n)$-minor has a separator of size $O(n/\ell + \ell h^2\log n)$ and that such a separator can be found in $O(mn/\ell)$ time. A time bound of $O(m + n^{2+\epsilon}/\ell)$ for any constant $\epsilon > 0$ was later given (W., FOCS'11) which is an improvement for non-sparse graphs. We give three new algorithms. The first has the same separator size and running time O(\mbox{poly}(h)\ell m^{1+\epsilon}). This is a significant improvement for small $h$ and $\ell$. If $\ell = \Omega(n^{\epsilon'})$ for an arbitrarily small chosen constant $\epsilon' > 0$, we get a time bound of O(\mbox{poly}(h)\ell n^{1+\epsilon}). The second algorithm achieves the same separator size (with a slightly larger polynomial dependency on $h$) and running time O(\mbox{poly}(h)(\sqrt\ell n^{1+\epsilon} + n^{2+\epsilon}/\ell^{3/2})) when $\ell = \Omega(n^{\epsilon'})$. Our third algorithm has running time O(\mbox{poly}(h)\sqrt\ell n^{1+\epsilon}) when $\ell = \Omega(n^{\epsilon'})$. It finds a separator of size O(n/\ell) + \tilde O(\mbox{poly}(h)\ell\sqrt n) which is no worse than previous bounds when $h$ is fixed and $\ell = \tilde O(n^{1/4})$. A main tool in obtaining our results is a novel application of a decremental approximate distance oracle of Roditty and Zwick.Comment: 16 pages. Full version of the paper that appeared at ICALP'14. Minor fixes regarding the time bounds such that these bounds hold also for non-sparse graph

A 7/9 - Approximation Algorithm for the Maximum Traveling Salesman Problem

We give a 7/9 - Approximation Algorithm for the Maximum Traveling Salesman Problem.Comment: 6 figure

Node-weighted Steiner tree and group Steiner tree in planar graphs

We improve the approximation ratios for two optimization problems in planar graphs. For node-weighted Steiner tree, a classical network-optimization problem, the best achievable approximation ratio in general graphs is Θ [theta] (logn), and nothing better was previously known for planar graphs. We give a constant-factor approximation for planar graphs. Our algorithm generalizes to allow as input any nontrivial minor-closed graph family, and also generalizes to address other optimization problems such as Steiner forest, prize-collecting Steiner tree, and network-formation games. The second problem we address is group Steiner tree: given a graph with edge weights and a collection of groups (subsets of nodes), find a minimum-weight connected subgraph that includes at least one node from each group. The best approximation ratio known in general graphs is O(log3 [superscript 3] n), or O(log2 [superscript 2] n) when the host graph is a tree. We obtain an O(log n polyloglog n) approximation algorithm for the special case where the graph is planar embedded and each group is the set of nodes on a face. We obtain the same approximation ratio for the minimum-weight tour that must visit each group

A separator theorem for string graphs and its applications

A string graph is the intersection graph of a collection of continuous arcs in the plane. We show that any string graph with in edges can be separated into two parts of roughly equal size by the removal of O(m(3/4)root log m) vertices. This result is then used to deduce that every string graph with n vertices and no complete bipartite subgraph K-t,K-t has at most c(t)n edges, where c(t) is a constant depending only on t. Another application shows that locally tree-like string graphs are globally tree-like: for any epsilon > 0, there is an integer g(epsilon) such that every string graph with n vertices and girth at least g(epsilon) has at most (1 + epsilon)n edges. Furthermore, the number of such labelled graphs is at most (1 + epsilon)(n) T(n), where T(n) = n(n-2) is the number of labelled trees on n vertices