Decomposition of Graphs into Induced Paths and Cycles I. Sahul Hamid, Abraham V. M. Abstract—A decomposition of a graph G is a collection ψ of subgraphs H1,H2,...,Hr of G such that every edge of G belongs to exactly one Hi . If each Hi is either an induced path or an induced cycle in G, then ψ is called an induced path decomposition of G. The minimum cardinality of an induced path decomposition of G is called the induced path decomposition number of G and is denoted by πi (G). In this paper we initiate a study of this parameter. Keywords—Path decomposition, Induced path decomposition, In- duced path decomposition number. I. I NTRODUCTION B Y a graph G =(V,E) we mean a finite, undirected graph with neither loops nor multiple edges. The order and size of G are denoted by n and m respectively. For graph theoretic terminology we refer to Chartrand and Lesniak [8]. All graphs in this paper are assumed to be connected and non-trivial. It is easy to see that If P = (v 0 ,v 1 ,...,v r ) is a path in a graph G, then v 1 ,v 2 ,..., v r−1 are called internal vertices of P and v 0 ,v r are called external vertices of P . If P =(v 0 ,v 1 ,...,v r ) and Q =(v r = w 0 ,w 1 ,...,w s ) are two paths in G, then the walk obtained by concatenating P and Q at v r is denoted by P ◦ Q and the path (v r ,v r−1 ,...,v 0 ) is denoted by P −1 . For a unicyclic graph G with cycle C, if w is a vertex of degree > 2 on C, then the maximal subtree T of G such that V (T ) ∩ V (C)= {w} is called the subtree rooted at w. A decomposition of a graph G is a collection of subgraphs H 1 ,H 2 ,...,H r of G such that every edge of G belongs to exactly one H i . Various types of decompositions and corresponding parameters have been studied by several authors by imposing conditions on the members of the decomposition. Some such decomposition parameters are path decomposi- tion number, acyclic path decomposition number and simple acyclic path decomposition number which are defined as follows. Let ψ = {H 1 ,H 2 ,...,H r } be a decomposition of a graph G. If each H i is either a path or a cycle, then ψ is called a path decomposition of G. If each H i is a path, then ψ is called an acyclic path decomposition of G. Further, an acyclic path decomposition in which any two paths have at most one vertex in common is called a simple acyclic path decomposition of G. The minimum cardinality of a path decomposition (acyclic path decomposition, simple acyclic path decomposition) of G is called the path decomposition number (acyclic path I. Sahul Hamid is working in Department of Mathematics, The Madura College, Madurai-11, India. (e-mail: sahulmat@yahoo.co.in) Abraham V. M. is working in Department of Mathematics, Christ Univer- sity, Bangalore, India. (e-mail: frabraham@christuniversity.in) Manuscript received ; revised decomposition number, simple acyclic path decomposition number) of G and is denoted by π(G)(π a (G),π as (G)). The parameter π a was introduced by Harary [9] and further studied by Harary and Schwenk [10], Peroche [11], Stanton et al. [12] and Arumugam and Suresh Suseela [7] who used the notation π for the acyclic path decomposition number of G and called an acyclic path decomposition as a path cover. The parameter π as was introduced by Arumugam and Sahul Hamid [5] who used π s for simple acyclic path decomposition number and called a simple acyclic path decomposition as a simple path cover and the parameter π was introduced by Arumugam et al. [6]. Further, by imposing on each of the decomposition defined above the condition that every vertex of G is an internal vertex of at most one member of the decomposition, we get another set of path covering parameters namely graphoidal covering number η(G), acyclic graphoidal covering number η a (G), simple graphoidal covering number η s (G) and simple acyclic graphoidal covering number η as (G) and all these parameters can be found respectively in [1], [7], [4] and [3]. Arumugam and Sahul Hamid [5] observed that every mem- ber of a simple acyclic path decomposition of a graph G is an induced path in G. However, a collection ψ of induced paths such that every edge of G is in exactly one path in ψ need not be a simple acyclic path decomposition of G. Motivated by this observation, Arumugam [2] introduced the concept of induced path decomposition and induced path decomposition number of a graph. In this paper we initiate a study of this parameter and determine the value of the parameter for several families of graphs. Also, we obtain some bounds and characterize the graphs attaining the bounds. II. MAIN RESULTS Definition 2.1. An induced path decomposition of a graph G is a path decomposition ψ of G such that every member of ψ is either an induced path or an induced cycle in G. The minimum cardinality of an induced path decomposition of G is called the induced path decomposition number of G and is denoted by π i (G). An induced path decomposition ψ of G with |ψ| = π i (G) is called a minimum induced path decomposition of G. Obviously, in a tree every path decomposition is an induced path decomposition. The following theorem says, in fact, that trees are the only graphs in which every path decomposition is induced. Theorem 2.2. Every path decomposition of G is an induced path decomposition of G if and only if G is a tree. World Academy of Science, Engineering and Technology International Journal of Mathematical and Computational Sciences Vol:3, No:11, 2009 939 International Scholarly and Scientific Research & Innovation 3(11) 2009 scholar.waset.org/1307-6892/3065 International Science Index, Mathematical and Computational Sciences Vol:3, No:11, 2009 waset.org/Publication/3065