Sparse square roots.

We show that it can be decided in polynomial time whether a graph of maximum degree 6 has a square root; if a square root exists, then our algorithm finds one with minimum number of edges. We also show that it is FPT to decide whether a connected n-vertex graph has a square root with at most n − 1 + k edges when this problem is parameterized by k. Finally, we give an exact exponential time algorithm for the problem of finding a square root with maximum number of edges

Algorithms for outerplanar graph roots and graph roots of pathwidth at most 2

Deciding whether a given graph has a square root is a classical problem that has been studied extensively both from graph theoretic and from algorithmic perspectives. The problem is NP-complete in general, and consequently substantial effort has been dedicated to deciding whether a given graph has a square root that belongs to a particular graph class. There are both polynomial-time solvable and NP-complete cases, depending on the graph class. We contribute with new results in this direction. Given an arbitrary input graph G, we give polynomial-time algorithms to decide whether G has an outerplanar square root, and whether G has a square root that is of pathwidth at most 2

Sparse Square Roots

We show that it can be decided in polynomial time whether a graph of maximum degree 6 has a square root; if a square root exists, then our algorithm finds one with minimum number of edges. We also show that it is FPT to decide whether a connected n-vertex graph has a square root with at most n − 1 + k edges when this problem is parameterized by k. Finally, we give an exact exponential time algorithm for the problem of finding a square root with maximum number of edges

Parameterized algorithms for finding square roots

We show that the following two problems are fixed-parameter tractable with parameter k: testing whether a connected n-vertex graph with m edges has a square root with at most n−1+k edges and testing whether such a graph has a square root with at least m−k edges. Our first result implies that squares of graphs obtained from trees by adding at most k edges can be recognized in polynomial time for every fixed k≥0; previously this result was known only for k=0. Our second result is equivalent to stating that deciding whether a graph can be modified into a square root of itself by at most k edge deletions is fixed-parameter tractable with parameter k

Computing square roots of graphs with low maximum degree

A graph H is a square root of a graph G if G can be obtained from H by adding an edge between any two vertices in H that are of distance 2. The Square Root problem is that of deciding whether a given graph admits a square root. This problem is known to be NP-complete for chordal graphs and polynomial-time solvable for non-trivial minor-closed graph classes and a very limited number of other graph classes. We prove that Square Root is O(n)-time solvable for graphs of maximum degree 5 and O(n4)-time solvable for graphs of maximum degree at most 6

Finding cactus roots in polynomial time

A graph H is a square root of a graph G, or equivalently, G is the square of H, if G can be obtained from H by adding an edge between any two vertices in H that are of distance 2. The SQUARE ROOT problem is that of deciding whether a given graph admits a square root. The problem of testing whether a graph admits a square root which belongs to some specified graph class H is called the H-SQUARE ROOT problem. By showing boundedness of treewidth we prove that SQUARE ROOT is polynomial-time solvable on some classes of graphs with small clique number and that H-SQUARE ROOT is polynomial-time solvable when H is the class of cactuses

Polynomial time recognition of squares of Ptolemaic graphs and 3-sun-free split graphs

The square of a graph G, denoted G(2), is obtained from G by putting an edge between two distinct vertices whenever their distance is two. Then G is called a square root of G(2). Deciding whether a given graph has a square root is known to be NP-complete, even if the root is required to be a chordal graph or even a split graph. We present a polynomial time algorithm that decides whether a given graph has a Ptolemaic square root. If such a root exists, our algorithm computes one with a minimum number of edges. In the second part of our paper, we give a characterization of the graphs that admit a 3-sun-free split square root. This characterization yields a polynomial time algorithm to decide whether a given graph has such a root, and if so, to compute one. (C) 2015 Elsevier B.V. All rights reserved

Finding cut-vertices in the square roots of a graph

The square of a given graph $H = (V, E)$ is obtained from $H$ by adding an edge between every two vertices at distance two in $H$. Given a graph class ${\cal H}$, the ${\cal H}$-Square Root problem asks for the recognition of the squares of graphs in ${\cal H}$. In this paper, we answer positively to an open question of [Golovach et al., IWOCA'16] by showing that the squares of cactus block graphs can be recognized in polynomial time. Our proof is based on new relationships between the decomposition of a graph by cut-vertices and the decomposition of its square by clique cutsets. More precisely, we prove that the closed neighbourhoods of cut-vertices in $H$ induce maximal prime complete subgraphs of $G = H^2$. Furthermore, based on this relationship, we introduce a quite complete machinery in order to compute from a given graph $G$ the block-cut tree of a desired square root (if any). Although the latter tree is not uniquely defined, we show surprisingly that it can only differ marginally between two different roots. Our approach not only gives the first polynomial-time algorithm for the ${\cal H}$-Square Root problem in different graph classes ${\cal H}$, but it also provides a unifying framework for the recognition of the squares of trees, block graphs and cactus graphs — among others