Complexity of Finding Graph Roots with Girth Conditions

Abstract

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 , however Motwani and Sudan proved that it is NP-complete to determine if a given graph G is the square of some graph H (of girth 3). In this paper we consider the characterization and recognition problems of graphs that are squares of graphs of small girth, i.e. to determine if G = H 2 for some graph H of small girth. The main results are the following.• There is a graph theoretical characterization for graphs that are squares of some graph of girth at least 7. A corollary is that if a graph G has a square root H of girth at least 7 then H is unique up to isomorphism.• There is a polynomial time algorithm to recognize if G = H 2 for some graph H of girth at least 6.• It is NP-complete to recognize if G = H 2 for some graph H of girth 4. These results almost provide a dichotomy theorem for the complexity of the recognition problem in terms of girth of the square roots. The algorithmic and graph theoretical results generalize previous results on tree square roots, and provide polynomial time algorithms to compute a graph square root of small girth if it exists. Some open questions and conjectures will also be discussed.

Finding Cactus Roots in Polynomial Time

“…This leads to the open problem of determining the complexity of H-SQUARE ROOT when H is the class of graphs of treewidth at most 2. We also Graphs of girth at least 4 [11] NP-complete…”

A linear kernel for finding square roots of almost planar graphs

“…Adamaszek and Adamaszek [1] proved that if a graph has a square root of girth at least 6, then this square root is unique up to isomorphism. Farzad, Lau, Le and Tuy [10] showed that recognizing graphs with a square root of girth at least g is polynomial-time solvable if g ≥ 6 and NP-complete if g = 4. The missing case g = 5 was shown to be NP-complete by Farzad and Karimi [9].…”

Algorithms for Outerplanar Graph Roots and Graph Roots of Pathwidth at Most 2

“…Significant advances have also been made on the complexity of H-Square Root. Previous results show that H-Square Root is polynomial-time solvable for the following graph classes H: trees [28], proper interval graphs [23], bipartite graphs [22], block graphs [26], strongly chordal split graphs [27], ptolemaic graphs [24], 3-sun-free split graphs [24], cactus graphs [18], cactus block graphs [12] and graphs with girth at least g for any fixed g ≥ 6 [14]. The result for 3-sun-free split graphs was extended to a number of other subclasses of split graphs in [25].…”

“…On the negative side, H-Square Root remains NP-complete for each of the following classes H: graphs of girth at least 5 [13], graphs of girth at least 4 [14], split graphs [23], and chordal graphs [23]. All known NP-hardness constructions involve dense graphs [13,14,23,31], and the square roots that occur in these constructions are dense as well. This, in combination with the aforementioned polynomial-time results, leads to our underlying research question:…”