TAILIEUCHUNG - The basis number of the semi-composition product of some graphs I

The basis number of a graph G is defined to be the least integer d such that there is a basis B of the cycle space of G such that each edge of G is contained in at most d members of B. We investigate the basis number of the semi-composition product of two paths and a cycle with a path. | Turk J Math 29 (2005) , 349 – 364. ¨ ITAK ˙ c TUB The Basis Number of the Semi-Composition Product of Some Graphs I M. M. M. Jaradat, E. A. Rawashdeh, M. Y. Alzoubi Abstract The basis number of a graph G is defined to be the least integer d such that there is a basis B of the cycle space of G such that each edge of G is contained in at most d members of B. We investigate the basis number of the semi-composition product of two paths and a cycle with a path. Key Words: Basis number; cycle space; fold; semi-composition product. 1. Introduction and Definitions Graph products have been the impetus for several areas of research. The revival of interest seems to be mostly due to the algorithmic point of view and how particular graphical parameters interact with graph products. In recent years, there was a growing literature on the basis number of graphs. Even more recently, the most attention has been given to the basis number of graphs obtained from graph products. We refer the readers to the papers [1], [2], [3], [4], [7], [8], [9]. All graphs under consideration are undirected, finite and simple. Our terminology and notations will be standard except as indicated. For undefined terms, see [5]. We use the symbols V (G) and E(G), respectively, to denote the vertex set and edge set of G. Given a graph G, let e1 , e2 , . . . , e

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