Induced Acyclic Path Decomposition in Graphs

A decomposition of a graph G is a collection ψ of graphs H1,H2, . . . , Hr of G such that every edge of G belongs to exactly one Hi. If each Hi is either an induced path in G, then ψ is called an induced acyclic path decomposition of G and if each Hi is a (induced) cycle in G then ψ is called a (induced) cycle decomposition of G. The minimum cardinality of an induced acyclic path decomposition of G is called the induced acyclic path decomposition number of G and is denoted by ¤Çia(G). Similarly the cyclic decomposition number ¤Çc(G) is defined. In this paper we begin an investigation of these parameters.




References:
[1] B. D. Acharya and E. Sampathkumar, Graphoidal covers and graphoidal
covering number of a graph, Indian J. Pure Appl. Math., 18(10) (1987),
882 - 890.
[2] S. Arumugam, Path covers in graphs, Lecture Notes of the National
Workshop on Decompositions of Graphs and Product Graphs held at
Annamalai University, Tamil Nadu, during January 3 - 7, 2006.
[3] S. Arumugam and I. Sahul Hamid, Simple acyclic graphoidal covers in
a graph, Australasian Journal of Combinatorics, 37 (2007), 243 - 255.
[4] S. Arumugam and I. Sahul Hamid, Simple Graphoidal Covers in a
Graph, Journal of Combin. Math. Combin. Comput., 64 (2008), 79 -
95.
[5] S. Arumugam and I. Sahul Hamid, Simple path covers in a graph,
International J. Math. Combin., 3 (2008), 94 - 104.
[6] S. Arumugam, I. Sahul Hamid and V. M. Abraham, Path decomposition
number of a graph, (submitted).
[7] S. Arumugam and J. Suresh Suseela, Acyclic graphoidal covers and path
partitions in a graph, Discrete Math., 190 (1998), 67 - 77.
[8] G. Chatrand and L. Lesniak, Graphs and Digraphs, Fourth Edition, CRC
Press, Boca Raton, 2004.
[9] F. Harary, Covering and Packing in graphs I, Ann. N. Y. Acad. Sci., 175
(1970), 198 - 205.
[10] F. Harary and A. J. Schwenk, Evolution of the path number of a graph,
covering and packing in graphs II, Graph Theory and Computing, Ed.
R. C. Road, Academic Press, New York, (1972), 39 - 45.
[11] B. Peroche, The path number of some multipartite graphs, Annals of
Discrete Math., 9 (1982), 193 - 197.
[12] I. Sahul Hamid and Abraham V. M., Decomposition of graphs into
induced paths and cycles, International Journal of Computational and
Mathematical Sciences, 3(7) (2009), 315 - 319.
[13] R. G. Stanton, D. D. Cowan and L. O. James, Some results on path
numbers, Proc. Louisiana Conf. on Combinatorics, Graph Theory and
Computing, (1970), 112 - 135.
[14] Juraj Bosak, Decompositions of Graphs, Kluwer Academic publishers,
Dorderecht, The Netherlands, (1990).