Finding Cactus Roots in Polynomial Time

Abstract

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.

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

“…To give a few examples, our results imply the aforementioned results for the cases where H is the class of forests [28] or cactus graphs (graphs in which every edge belongs to at most one cycle) [18], which both are subclasses of outerplanar graphs that satisfy conditions (i) an (ii). To give another example, a connected graph has pathwidth 1 if and only if it is a caterpillar (a tree which can be modified in a path after removing all vertices of degree 1).…”

“…The goal is to obtain a graph whose treewidth is bounded by a constant, which enables us to solve the problem in polynomial time after expressing it in Monadic Second-Order Logic and applying a classical result of Courcelle [9]. This idea has been used before (see, for instance, [7,8,18,19]), but in this paper we formalize the idea into a general framework. We discuss this framework in detail in Section 3.…”

A linear kernel for finding square roots of almost planar graphs

“…Lin and Skiena [23] gave a linear-time algorithm for recognizing squares of trees; they also proved that Square Root can be solved in linear time for planar graphs. Le and Tuy [21] generalized the above results for trees [23,29] to block graphs, whereas we recently gave a polynomial-time algorithm for recognizing squares of cactuses [11]. Nestoridis and Thilikos [28] proved that Square Root is not only polynomial-time solvable for the class of planar graphs but for any non-trivial minor-closed graph class, that is, for any graph class that does not contain all graphs and that is closed under taking vertex deletions, edge deletions and edge contractions.…”

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

“…It is known that Square Root is polynomial-time solvable on planar graphs [28], and more generally, on every non-trivial minor-closed graph class [33]. Polynomial-time algorithms also exist if the input graph G belongs to one of the following graph classes: block graphs [26], line graphs [29], trivially perfect graphs [30], threshold graphs [30], graphs of maximum degree 6 [7], graphs of maximum average degree smaller than 46 11 [18] 1 graphs with clique number at most 3 [18], and graphs with bounded clique number and no long induced path [18]. On the negative side, Square Root is NP-complete on chordal graphs [23].…”

“…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].…”