Semi-Image Neighborhood Block Graphs with Crossing Numbers

The advent of graph theory has played a prominent r ole in wide variety of engineering applications. Minimization of crossing numbers in g raphs optimizes its use in many applications. In this paper, we establish the neces sary and sufficient condition for SemiImage neighborhood block graph to have crossing num ber 3. We also prove that the SemiImage neighborhood block graph NBI(G) of a graph never has crossing numbers k, where 1 ≤ k ≤ 6.


Introduction
Crossing number minimization is one of the fundamental optimization problems because it is related to various other widely used notions.Besides its mathematical interest, there are numerous applications, most notably those in VLSI design and in combinatorial geometry [1].Researchers in computer science are focusing their attention to this area of graph theory as the study of crossing numbers of graphs finds applications in network design, higraph system model and circuit layout.Minimizing the number of wire crossings in a circuit greatly reduces the chance of cross-talk in long crossing wires carrying the same signal and also allows for faster operation and less power dissipation.It is also an important measure of non-planarity of a graph.
Let G = (V, E) be a simple finite, undirected and without loops or multiple lines graph with vertex set V and edge set E. A drawing D of a graph G is a representation of G in the Euclidean plane R 2 where vertices are represented as distinct points and edges by simple polygonal arcs joining points that correspond to their end vertices.A drawing D is good or clean, if no edge crosses itself, no pair of adjacent edges cross, two edges cross at most once and no more than two edges cross at one point.The number of crossings of D is denoted by Cr(D) and is called the crossing number of the drawing D. The crossing number Cr(G) of a graph G is the minimum Cr(D) taken from over all good or clean drawings D of G.If a graph G admits a drawing D with Cr(D) =0 then G is said to be planar; otherwise non-planar.It is well known that K 5 , the complete graph on five vertices and K 3,3 the complete bipartite graph with three vertices in its classes are non-planar.According to Kuratowski's famous theorem, a graph is planar if and only if it contains no subdivision of K 5 or K 3,3 [2,3].The study of crossing numbers began during the Second World War with Paul Turan.The bounds for Cr(K n ) and Cr(K m,n ) are obtained by Guy [4].The Semi-Image neighborhood block graph NB I (G) of a graph G is defined as the graph that has the point set V(G) U V ' (G) U b(G).Two points of V (G) and V ' (G) are adjacent, if they are adjacent in G. Two points V i and V ' j are adjacent, if i=j as well as two points of V (G) are incident on b(G).This concept is introduced in ref. [5].The crossing Number problem for generalized Peteresen graph has been investigated in ref. [6].
The following theorems are used to prove our main results.
Theorem A [5]: The Semi-Image neighborhood block graph NB I (G) of a graph G is planar if and only if G is outer planar.
Theorem B [5]: The Semi-Image neighborhood block graph NB I (G) of a graph G is outer planar if and only if every component of G is a path.
Proof: Suppose G is K 4 or K 2, 3 .then, we can easily say that the graphs NB I (K 4 ) and NB I (K 2, 3 ) are non-planar and drawings in Fig. 1 and Fig. 2 show that the crossing number is at most three.One may also observe that NB I (K 4 ) and NB I (K 2, 3 ) can not be drawn without crossing K 4 or K 2, 3 line.However, Fig. 3 shows that the deletion of such a point results a non-planar graph (it contains a homeomorphism of K 3,3 ).Therefore, the crossing number must be equal to three.□ Fig. 3. Semi-Image neighborhood block graph G with crossing number three which is a homeomorphism of K 3, 3 .
In fact, we can also prove that for any graph G, the semi-image neighborhood block graph NB I (G) of a graph G, never has crossing number k; where 1 ≤k ≤ 6 (i.e.k=1,2,4,5 or 6).
From the above cases, it can be concluded that G is K 4 or K 2, 3 .
□ Next, we can also prove that the Semi-Image neighborhood block graph of any nonplanar graph has crossing number greater than six.

Theorem3:
The Semi-Image neighborhood block graph of any non-planar graph has crossing number at least seven.
Proof: By Kuratowski's theorem on planar graphs, it is sufficient to prove that NB I (K 5 ) and NB I (K 3, 3 ) have crossing number at least seven.The graphs NB I (K 3, 3 ) and NB I (K 5 ) are isomorphic to K 2 x K 3,3 and K 2 x K 5 .If NB I (K 5 ) and NB I (K 3,3 ) have crossing number less than seven, then the deletion of a few pairs of its points must result in a non-planar graph with at least two crossings.But, Theorem 1 and Theorem 2 imply that any graph obtained by removing a point from K 5 or K 3, 3 has a non-planar semi-image neighborhood block graph with at least three crossings.Hence, the Semi-Image neighborhood block graph of both K 5 and K 3, 3 has at least seven crossings in each drawing.

Conclusion
In this paper, the necessary and sufficient condition is established for Semi-Image neighborhood block graph to have crossing number three.The proof of "Semi-Image neighborhood block graph NB I (G) of a graph never has crossing numbers k, where 1 ≤ k ≤ 6" is also given.Finally, the necessary and sufficient conditions are provided for semi-image neighborhood block graph NB I (G) of a graph to have forbidden sub graphs for crossing numbers k (k = 1, 2 or 3).