BIRB796 In Galeana S nchez and Delgado

In  [6] Galeana-Sánchez and Delgado-Escalante used the work of Arpin and Linek  [5] in order to introduce the following concepts:

Since the existence of an -walk between two vertices does not guarantee the existence of an -path between those vertices and the concatenation of two -paths is not always an -path, we can claim that if has an -kernel by walks, then not necessarily has an -kernel as the example in Fig. 1 shows. In Fig. 1 we have that is an -kernel by walks of , because (, , , , ) is an -walk in that finishes in and it BIRB796 contains every vertex of . It is easy to check that has no -kernel (notice that every -independent set of has cardinality one).
We also claim that if has an -kernel, then not necessarily has an -kernel by walks as the example in Fig. 2 shows. In Fig. 2 we have that is an -kernel in . It is easy to see that has no -kernel by walks (notice that every -independent set by walks in has cardinality one because (, , , , ) is an -walk between and in ).
In  [6] Galeana-Sánchez and Delgado-Escalante proved the existence of -kernels in possibly infinite -colored digraphs. In  [7] Galeana-Sánchez and Sánchez-López showed necessary and sufficient conditions for the existence of -kernels in the -join of digraphs. Finally in  [8] Galeana-Sánchez and Sánchez-López showed more conditions for the existence of -kernels in infinite digraphs.

Main results
The following lemma, which was proved in  [5], will be useful in order to prove Theorems 1.5 and 1.6.

Theorem 3.2 allows us to establish the following results. Before we need a definition.
Let be a digraph with , , and a sequence of vertex disjoint digraphs with and for each . The -join of the digraph and the sequence is the digraph () such that:
Notice that from the definition of () we have that () contains an isomorphic digraph to for each . Denote by the copy of in ().
Observe that .
The following theorem shows how to produce more digraphs in from a digraph in . It is necessary to mention that the following result was proved in  [5] by Arpin and Linek. Here we are going to prove that Theorem 3.4 is also a direct consequence of Theorem 3.2.

The following result shows another sufficient condition for the existence of an -kernel by walks.

Acknowledgments

Introduction
A five-connected planar graph is called a doughnut graph if has an embedding such that (a) has two vertex-disjoint faces each of which has exactly vertices, , and all the other faces of has exactly three vertices; and (b) has the minimum number of vertices satisfying condition (a). Fig. 1(a) illustrates a doughnut graph where and are two vertex disjoint faces. Faces and are depicted by thick lines. The name of doughnut graph was chosen in  [1] for such a graph since the graph has a doughnut like embedding, as illustrated in Fig. 1(b). The class of doughnut graphs is an interesting class of graphs which was recently introduced in graph drawing literature for it’s beautiful area-efficient drawing properties  [1–3]. A doughnut graph admits a straight-line grid drawing with linear area  [1,3]. Any spanning subgraph of a doughnut graph also admits straight-line grid drawing with linear area  [2,3]. The outerplanarity of this class is 3  [3].
Given a graph , natural numbers , , , such that , we wish to find a -partition of the vertex set such that and induces a connected subgraph of for each . The problem of finding a -partition of a given graph often appears in the load distribution among different power plants and the fault-tolerant routing of communication networks  [4,5]. A doughnut graph is -partitionable  [6].
A class of graph has recursive structure if every instance of it can be created by connecting the smaller instances of the same class of graphs. In this paper, we show that any instance of a doughnut graph can be constructed by connecting smaller instances of doughnut graphs. We show that one can find a shortest path between any pair of vertices and of a doughnut graph in time where is the length of shortest path between and by exploiting its beautiful structure. We study the other topological properties like degree, diameter, connectivity and fault tolerance. We show that it’s diameter is . It has maximal fault tolerance, and has ring embedding since it is Hamilton-connected. One may explore the suitability of a doughnut graph as an interconnection network since some of its properties are similar to that of the graph classes usually used for interconnection networks.