Optimality program in segment and string graphs

Planar graphs are known to allow subexponential algorithms running in time $2^{O(\sqrt n)}$ or $2^{O(\sqrt n \log n)}$ for most of the paradigmatic problems, while the brute-force time $2^{\Theta(n)}$ is very likely to be asymptotically best on general graphs. Intrigued by an algorithm packing curves in $2^{O(n^{2/3}\log n)}$ by Fox and Pach [SODA'11], we investigate which problems have subexponential algorithms on the intersection graphs of curves (string graphs) or segments (segment intersection graphs) and which problems have no such algorithms under the ETH (Exponential Time Hypothesis). Among our results, we show that, quite surprisingly, 3-Coloring can also be solved in time $2^{O(n^{2/3}\log^{O(1)}n)}$ on string graphs while an algorithm running in time $2^{o(n)}$ for 4-Coloring even on axis-parallel segments (of unbounded length) would disprove the ETH. For 4-Coloring of unit segments, we show a weaker ETH lower bound of $2^{o(n^{2/3})}$ which exploits the celebrated Erd\H{o}s-Szekeres theorem. The subexponential running time also carries over to Min Feedback Vertex Set but not to Min Dominating Set and Min Independent Dominating Set.Comment: 19 pages, 15 figure

The price of connectivity for cycle transversals

For a family of graphs F, an F-transversal of a graph G is a subset S⊆V(G) that intersects every subset of V(G) that induces a subgraph isomorphic to a graph in F. Let tF(G) be the minimum size of an F-transversal of G, and View the MathML source be the minimum size of an F-transversal of G that induces a connected graph. For a class of connected graphs G, we say that the price of connectivity of F-transversals is multiplicative if, for all G∈G, View the MathML source is bounded by a constant, and additive if View the MathML source is bounded by a constant. The price of connectivity is identical if tF(G) and View the MathML source are always equal and unbounded if View the MathML source cannot be bounded in terms of tF(G). We study classes of graphs characterized by one forbidden induced subgraph H and F-transversals where F contains an infinite number of cycles and, possibly, also one or more anticycles or short paths. We determine exactly those classes of connected H-free graphs where the price of connectivity of these F-transversals is unbounded, multiplicative, additive, or identical. In particular, our tetrachotomies extend known results for the case when F is the family of all cycles