3 research outputs found
Large-girth roots of graphs
We study the problem of recognizing graph powers and computing roots of
graphs. We provide a polynomial time recognition algorithm for r-th powers of
graphs of girth at least 2r+3, thus improving a bound conjectured by Farzad et
al. (STACS 2009). Our algorithm also finds all r-th roots of a given graph that
have girth at least 2r+3 and no degree one vertices, which is a step towards a
recent conjecture of Levenshtein that such root should be unique. On the
negative side, we prove that recognition becomes an NP-complete problem when
the bound on girth is about twice smaller. Similar results have so far only
been attempted for r=2,3.Comment: 14 pages, 4 figure
Graph Powers: Hardness Results, Good Characterizations and Efficient Algorithms
Given a graph H = (V_H,E_H) and a positive integer k, the k-th power of H, written H^k, is the graph obtained from H by adding edges between any pair of vertices at distance at most k in H; formally, H^k = (V_H, {xy | 1 <= d_H (x, y) <= k}). A graph G is the k-th power of a graph H if G = H^k, and in this case, H is a k-th root of G. Our investigations deal with the computational complexity of recognizing k-th powers of general graphs as well as restricted graphs. This work provides new NP-completeness results, good characterizations and efficient algorithms for graph powers