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 research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP/2007(FP/ -2013/ERC Grant Agreement n. 267959. The research also has been supported by EPSRC (EP/G043434/1) and ANR Blanc AGAPE (ANR-09-BLAN-0159-03). A preliminary version of this paper appeared as an extended abstract in the proceedings of WG 2013 [4].
Citation for published item:gohefertD wF nd gouturierD tEpF nd qolovhD FeF nd urtshD hF nd ulusmD hF @PHIQA 9prse squre rootsF9D in qrphEtheoreti onepts in omputer siene X QWth snterntionl orkshopD q PHIQD v¤ uekD qermnyD IWEPI tune PHIQ Y revised ppersF ferlinD reidelergX pringerD ppF IUUEIVVF veture notes in omputer sieneF @VITSAF Further information on publisher's website:Publisher's copyright statement:The nal publication is available at Springer via http://dx.doi.org/10.1007/978-3-642-45043-316Additional information: 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. Abstract. We show that it can be decided in polynomial time whether a graph of maximum degree 6 has a square root; if a square root exists, then our algorithm finds one with minimum number of edges. We also show that it is FPT to decide whether a connected n-vertex graph has a square root with at most n − 1 + k edges when this problem is parameterized by k. Finally, we give an exact exponential time algorithm for the problem of finding a square root with maximum number of edges.
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.
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