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 NPcomplete 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.
\begin{itemize}
\item 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.
\item There is a polynomial time algorithm to recognize if $G=H^2$ for some graph $H$ of girth at least $6$.
\item It is NPcomplete to recognize if $G=H^2$ for some graph $H$ of girth $4$.
\end{itemize}
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.
BibTeX  Entry
@InProceedings{farzad_et_al:LIPIcs:2009:1827,
author = {Babak Farzad and Lap Chi Lau and Van Bang Le and Nguyen Ngoc Tuy},
title = {{Computing Graph Roots Without Short Cycles}},
booktitle = {26th International Symposium on Theoretical Aspects of Computer Science},
pages = {397408},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783939897095},
ISSN = {18688969},
year = {2009},
volume = {3},
editor = {Susanne Albers and JeanYves Marion},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2009/1827},
URN = {urn:nbn:de:0030drops18279},
doi = {10.4230/LIPIcs.STACS.2009.1827},
annote = {Keywords: Graph roots, Graph powers, Recognition algorithms, NPcompleteness}
}
Keywords: 

Graph roots, Graph powers, Recognition algorithms, NPcompleteness 
Collection: 

26th International Symposium on Theoretical Aspects of Computer Science 
Issue Date: 

2009 
Date of publication: 

19.02.2009 