Square-Root Finding Problem In Graphs, A Complete Dichotomy Theorem

Graph G is the square of graph H if two vertices x,y have an edge in G if and only if x,y are of distance at most two in H. Given H it is easy to compute its square H^2. Determining if a given graph G is the square of some graph is not easy in general. Motwani and Sudan proved that it is NP-complete to determine if a given graph G is the square of some graph. The graph introduced in their reduction is a graph that contains many triangles and is relatively dense. Farzad et al. proved the NP-completeness for finding a square root for girth 4 while they gave a polynomial time algorithm for computing a square root of girth at least six. Adamaszek and Adamaszek proved that if a graph has a square root of girth six then this square root is unique up to isomorphism. In this paper we consider the characterization and recognition problem of graphs that are square of graphs of girth at least five. We introduce a family of graphs with exponentially many non-isomorphic square roots, and as the main result of this paper we prove that the square root finding problem is NP-complete for square roots of girth five. This proof is providing the complete dichotomy theorem for square root problem in terms of the girth of the square roots

EDGE IDEALS OF SQUARES OF TREES

We describe all the trees with the property that the corresponding edge ideal of the square of the tree has a linear resolution. As a consequence, we give a complete characterization of those trees T for which the square is co-chordal, that is the complement of the square, (T2)c, is a chordal graph. For particular classes of trees such as paths and double brooms, we determine the Krull dimension and the projective dimension

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

The Structure Of Functional Graphs For Functions From A Finite Domain To Itself For Which A Half Iterate Exists

The notion of a replica of a nontrivial in-tree is defined. A result enabling to determine whether an in-tree is a replica of another in-tree employing an injective mapping between some subsets of sources of these in-trees is presented. There are given necessary and sufficient conditions for the existence of a functional square root of a function from a finite set to itself through presenting necessary and sufficient conditions for the existence of a square root of a component of the functional graph for the function and for the existence of a square root of the union of two components of the functional graph for the function containing cycles of the same length using the concept of the replica

Uniqueness of graph square roots of girth six

We prove that if two graphs of girth atleast 6 have isomorphic squares, then the graphs themselves are isomorphic. This is the best possible extension of the results of Ross and Harary on trees and the results of Farzad et al. on graphs of girth at least 7. We also make a remark on reconstruction of graphs from their higher powers