Computing square roots of graphs with low maximum degree

Abstract

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(n 4)-time solvable for graphs of maximum degree at most 6.

Finding Cactus Roots in Polynomial Time

“…Our motivation for doing so comes from the observation that SQUARE ROOT is readily seen to be polynomial-time solvable for graphs with clique number at most 3 (the only square roots a connected graph on n vertices with clique number 3 may have are the cycle or path on n vertices). Moreover, by identifying such classes of graphs, our results complement existing polynomial-time results for other classes of graphs with a small clique number, such as planar graphs [23] and graphs of maximum degree 6 [4]. We prove that SQUARE ROOT is polynomial-time solvable for the classes of 3-degenerate graphs and (K r , P t )-free graphs by showing that squares in these two graph classes have bounded treewidth.…”

“…We also need to define the following more general variant introduced in [4] for general square roots:…”

“…Our research is motivated by the result of Motwani and Sudan [26] who proved in 1994 that SQUARE ROOT is NP-complete. Afterwards, SQUARE ROOT was shown to be polynomial-time solvable for various graph classes, such as planar graphs [23], or more generally, any non-trivial minor-closed graph class [28]; block graphs [21]; line graphs [24]; trivially perfect graphs [25]; threshold graphs [25]; graphs of maximum degree 6 [4] and graphs of maximum average degree smaller than 46 11 [14]. It was also shown that SQUARE ROOT is NP-complete for chordal graphs [18].…”

“…It was also shown that SQUARE ROOT is NP-complete for chordal graphs [18]. We refer to [4,5,14] for a number of parameterized complexity results on SQUARE ROOT.…”

“…As such we may use Courcelle's Theorem [6] to obtain an alternative (but comparable) proof for the polynomial-time result of Lin and Skiena [23] for SQUARE ROOT restricted to planar graphs. We note that the polynomial-time algorithms for solving SQUARE ROOT for graphs of maximum degree at most 6 [4] and graphs of maximum average degree less than 46 11 [14] are also based on showing that the graphs which permit square roots also have bounded treewidth. Nestoridis and Thilikos [28] proved their result for minorclosed graph classes by showing boundedness of carving width.…”