# The Square Root Phenomenon in Planar Graphs

## Abstract

Most of the classical NP-hard problems remain NP-hard when restricted to planar graphs, and only exponential-time algorithms are known for the exact solution of these planar problems. However, in many cases, the exponential-time algorithms on planar graphs are significantly faster than the algorithms for general graphs: for example, 3-Coloring can be solved in time \(2^{O(\sqrt{n})}\) in an *n*-vertex planar graph, whereas only 2^{ O(n)}-time algorithms are known for general graphs. For various planar problems, we often see a square root appearing in the running time of the best algorithms, e.g., the running time is often of the form \(2^{O(\sqrt{n})}\), \(n^{O(\sqrt{k})}\), or \(2^{O(\sqrt{k})}\cdot n\). By now, we have a good understanding of why this square root appears. On the algorithmic side, most of these algorithms rely on the notion of treewidth and its relation to grid minors in planar graphs (but sometimes this connection is not obvious and takes some work to exploit). On the lower bound side, under a complexity assumption called Exponential Time Hypothesis (ETH), we can show that these algorithms are essentially best possible, and therefore the square root has to appear in the running time.