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.

A graph H is a square root of a graph G if G can be obtained from H by the addition of edges between any two vertices in H that are of distance 2 from each other. The Square Root problem is that of deciding whether a given graph admits a square root. We consider this problem for planar graphs in the context of the "distance from triviality" framework. For an integer k, a planar+kv graph (or k-apex graph) is a graph that can be made planar by the removal of at most k vertices. We prove that a generalization of Square Root, in which some edges are prescribed to be either in or out of any solution, has a kernel of size O(k) for planar+kv graphs, when parameterized by k. Our result is based on a new edge reduction rule which, as we shall also show, has a wider applicability for the Square Root problem.

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.

The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-pro t purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. Abstract. A homomorphism from a graph G to a graph H is a vertex mapping f from the vertex set of G to the vertex set of H such that there is an edge between vertices f (u) and f (v) of H whenever there is an edge between vertices u and v of G. The H-COLOURING problem is to decide if a graph G allows a homomorphism to a fixed graph H. We continue a study on a variant of this problem, namely the SURJECTIVE H-COLOURING problem, which imposes the homomorphism to be vertex-surjective. We build upon previous results and show that this problem is NP-complete for every connected graph H that has exactly two vertices with a self-loop as long as these two vertices are not adjacent. As a result, we can classify the computational complexity of SURJECTIVE H-COLOURING for every graph H on at most four vertices.

Citation for published item:qolovhD etr nd urtshD hieter nd ulusmD hni¤ el nd tewrtD enthony @PHITA 9qures of low lique numerF9D iletroni notes in disrete mthemtisFD SS F ppF IWSEIWVF Further information on publisher's website: Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. AbstractThe Square Root problem is that of deciding whether a given graph admits a square root. This problem is only 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. By researching boundedness of the treewidth of a graph, we prove that Square Root is polynomial-time solvable on various graph classes of low clique number that are not minor-closed.