TY - GEN
T1 - Results on independent sets in categorical products of graphs, the ultimate categorical independence ratio and the ultimate categorical independent domination ratio
AU - Hon, Wing Kai
AU - Kloks, Ton
AU - Liu, Ching Hao
AU - Liu, Hsiang Hsuan
AU - Poon, Sheung Hung
AU - Wang, Yue Li
N1 - Funding Information:
Supported in part by grants 100-2628-E-007-020-MY3 and 101-2218-E-007-001 of the National Science Council (NSC), Taiwan, R.O.C.
PY - 2014
Y1 - 2014
N2 - We first present polynomial algorithms to compute maximum independent sets in the categorical products of two cographs or two splitgraphs, respectively. Then we prove that computing the independent set of the categorical product of a planar graph of maximal degree three and K 4 is NP-complete. The ultimate categorical independence ratio of a graph G is defined as lim k → ∞ α(G k )/n k . The ultimate categorical independence ratio can be computed in polynomial time for cographs, permutation graphs, interval graphs, graphs of bounded treewidth and splitgraphs. Also, we present an O â̂- (3 n/3) exact, exponential algorithm for the ultimate categorical independence ratio of general graphs. We further present a PTAS for the ultimate categorical independence ratio of planar graphs. Lastly, we show that the ultimate categorical independent domination ratio for complete multipartite graphs is zero, except when the graph is complete bipartite with color classes of equal size (in which case it is 1/2).
AB - We first present polynomial algorithms to compute maximum independent sets in the categorical products of two cographs or two splitgraphs, respectively. Then we prove that computing the independent set of the categorical product of a planar graph of maximal degree three and K 4 is NP-complete. The ultimate categorical independence ratio of a graph G is defined as lim k → ∞ α(G k )/n k . The ultimate categorical independence ratio can be computed in polynomial time for cographs, permutation graphs, interval graphs, graphs of bounded treewidth and splitgraphs. Also, we present an O â̂- (3 n/3) exact, exponential algorithm for the ultimate categorical independence ratio of general graphs. We further present a PTAS for the ultimate categorical independence ratio of planar graphs. Lastly, we show that the ultimate categorical independent domination ratio for complete multipartite graphs is zero, except when the graph is complete bipartite with color classes of equal size (in which case it is 1/2).
UR - http://www.scopus.com/inward/record.url?scp=84958554420&partnerID=8YFLogxK
U2 - 10.1007/978-3-319-04657-0_23
DO - 10.1007/978-3-319-04657-0_23
M3 - Conference contribution
AN - SCOPUS:84958554420
SN - 9783319046563
T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
SP - 237
EP - 248
BT - Algorithms and Computation - 8th International Workshop, WALCOM 2014, Proceedings
PB - Springer Verlag
T2 - 8th International Workshop on Algorithms and Computation, WALCOM 2014
Y2 - 13 February 2014 through 15 February 2014
ER -