Embedding Sun Graphs in a Single page

A book consists of a line in the 3-dimensional space, called the spine, and a number of pages, each a half-plane with the spine as boundary. A book embedding (, ) of a graph consists of a linear ordering of , of vertices, called the spine ordering, along the spine of a book and an assignment , of edges to pages so that edges assigned to the same page can be drawn on that page without crossing. That is, we cannot find vertices u, v, x, y with (u) < (x) < (v) < (y), yet the edges uv and xy are assigned to the same page, that is (uv) = (xy). The book thickness or page number of a graph G is the minimum number of pages in required to embed G in a book. In this paper we consider the Sun Graph or the Trampoline graph and obtain the printing cycle for embedding the Sun Graph in a single page. We also give a linear time algorithm for such an embedding.


Introduction
The growth of the subject 'graph theory' has been very rapid in recent years, particularly since the domain of its application is extremely varied. Graph algorithms play a very important role in design of various computer networks. Among theproblems one comes across in graph theory, is the embedding of graphs. A particular way of embedding graphs is in the pages of a book. The book embedding of graphs was first introduced by Bernhart and Kainen [1] and since then, many researchers have actively studied it. Determining the book thickness for general graphs is NPhard. But obtaining the book thickness for particular graphs have been found to be possible. The book embeddings have been studied for many classes of graphs. To name a few, we have: Complete Graphs [1,2], Complete Bipartite Graphs [10], Trees, Grids and X-trees [3], hypercubes [3,9], incomplete hypercubes [8], iterated line digraphs [6], de Bruijn graphs, Kautz graphs, shuffle-exchange graphs [7], for each of which embedding in books have been studied.
The book embedding problem has many different applications, which include sorting with parallel stacks, single-row routing of printed circuit boards, and the design of fault-tolerant processor arrays [4,11].

Preliminaries
Definition: The Sun Graph (or the trampoline graph) network of order n denoted by S n is defined as follows. S 1 is K 3 . S n , n ≥ 2 is obtained from S n 1 by adding a new vertex v corresponding to each edge ab on the Hamiltonian cycle of S n and including the edges (v, a) and (v, b). See Figure 1. We give the definitions which are required for the discussion.

Embedding the Sun graph of order k in a single page
We shall discuss the embedding of the Sun graph of order k.

Lemma 1:
The Sun graph S k , k  1 can be embedded in a single page.

Proof:
The Sun graph S k , k  1 is outerplanar [3]. Hence it can be embedded in a single page. Hence the lemma.
We shall now find the printing cycle of the book embedding of the Sun graph in a single page.

Theorem 2:
The printing cycle S n is  n 1 (1, 2, 3),n  1embeds S n , n  1 in a single page.
Proof: Let us prove the theorem by induction on n, the order of the Sun graph. When n = 1, S 1 is Let  k 1 (1, 2, 3),...a, 1, a 2 3.2 k 1 be the hamilton cycle in s k . To obtain s k 1 , we introduce vertices so that is the vertex set of a K 3 graph, Apparently, this printing cycle would embed s k 1 in a single page. This completes the induction and the theorem is proved.

Embedding Algorithm
We now give the algorithm to obtain the printing cycle for embedding the sun graph s k in a single page.  The 'if' statement in line 9 singles out the case i = 1, for which the printing cycle has already been initialized in line 6. Line 26 increments the value if i to obtain the printing cycle of the next S i or to exit the loop when i has reached the value k+1.

Proof of correctness of the embedding algorithm
We now give the proof of correctness of the Algorithm BKEMBEDSUN(a,k). The number of terms involved in the sequence  r 2 (1, 2, 3) is 3  2 r 2 , the present value of m and l is assigned this value in line 11. Thus l will contain the number of vertices of S r 1 . In line 12, m is doubled so that its value will be the 2  3  2 r 2  3 2 r 1 , the number of vertices in S r . (1, 2, 3) , the printing cycle of S r .
Termination: When i reaches the value k+1, the while( ) loop exits and a[1, ..., m] will contain the printing cycle obtained at the end of the previous iteration corresponding to i = k, which is  k 1 (1, 2, 3) , the printing cycle of S k .
Hence Algorithm BKEMBEDSUN(a, k) will produce  k 1 (1, 2, 3) , the printing cycle of the Sun graph S k . Hence the theorem is proved.

Time Complexity
The following theorem proves that Algorithm BKEMBEDSUN(a, k)

Conclusion:
From the above discussion, it is apparent that even for outerplanar graphs, it may be very difficult to obtain the printing cycle that would embed such graphs in a single page. This once again underlines the fact that not many general results exist or are obtained in the book embedding of graphs.