The Square Root Phenomenon in Planar Graphs

A Linear Kernel for Finding Square Roots of Almost Planar Graphs

A graph H is a square root of a graph G if G can be obtained from H by the addition of edges between any two vertices in H that are of distance 2 from each other. The Square Root problem is that of deciding whether a given graph admits a square root. We consider this problem for planar graphs in the context of the “distance from triviality” framework. For an integer k, a planar+kv graph is a gr...

Algorithms for Square Roots of Graphs

The n-th power (n 1) of a graph G = (V; E), written G n , is deened to be the graph having V as its vertex set with two vertices u; v adjacent in G n if and only if there exists a path of length at most n between them. Similarly, graph H has an n-th root G if G n = H. For the case of n = 2, we say that G 2 is the square of G and G is the square root of G 2. Here we give a linear time algorithm ...

On subexponential parameterized algorithms for Steiner Tree and Directed Subset TSP on planar graphs

There are numerous examples of the so-called “square root phenomenon” in the field of parameterized algorithms: many of the most fundamental graph problems, parameterized by some natural parameter k, become significantly simpler when restricted to planar graphs and in particular the best possible running time is exponential in O( √ k) instead of O(k) (modulo standard complexity assumptions). We...

C~i~~h. Et K I Kl

We give an algorithm fr the Computation of K-terminal reliability in planar graphs, whose worst-case complexity is strictly exponential in the square root of the total number of

Planar graphs with square or cube root are four-colorable