The Block-transformation Graph ۵
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(2) 223. B. Basavanagoud, et al., J. Comp. & Math. Sci. Vol.6 (4), 222-227 (2015). planar graph if G can be drawn in a plane so that no two of its edges cross each other. A graph that is not planar is called nonplanar. A set of vertices of a planar graph G is called an inner vertex set S of G if G can be drawn on the plane in such a way that each vertex of S lies only on the interior region and S contains the minimum possible number of vertices of G. The number of vertices of S is said to be the inner vertex number of G and is denoted by i(G). A graph G is said to be k-minimally nonouterplanar if i(G) = k, k ≥ 1. A 1-minimally nonouterplanar graph is called minimally nonouterplanar. These concepts are introduced in8. The crossing number Cr(G) of a graph G is the minimum number of pairwise intersections of its edges when G is drawn in the plane. Obviously, Cr(G) = 0 if and only if G is planar. A graph G has crossing number 1 if and only if Cr(G) = 1, (see in5). A graph G is called a theta wheel if G is a cycle C together with a new vertex adjacent with exactly two nonadjacent vertices of C or adjacent with at least 3 vertices of C. A minimallynonouterplanar graph G is called theta-minimally nonouterplanar if G has a theta wheel such that at least one edge of a theta wheel is a boundary edge. If G and H are the graphs with property that the identification of any vertex of G with an arbitrary vertex of H results in a unique graph (up to isomorphism), then we write G ⋅ H for this graph. If B is a block of G with its vertex set,
(3) , , … , ; 2 then we say that , 1 , and B are incident with each other. If two blocks and of G have a common cutvertex, then we say that and are adjacent blocks of G. Let U(G) denote the block set of G and is the set { : is a block of G }. In7 the concept of total-block graph was introduced. The total-block graph of a graph G is the graph whose set of vertices is the union of set of vertices and set of blocks of G in which two vertices are adjacent if and only if the corresponding members of G are adjacent or incident. B. Basavanagoud et al.1 introduced some new graphical transformations which generalize the concept of total-block graph. Let ! " , # be a graph, and x, y be two vertices of " $ % . We define the associativity of x and y is 1 if they are adjacent or incident, and 0 otherwise. Let αβγ be a 3-permutation of the set {1, 0}. We say that x and y correspond to the first term α(resp. the second term β or the third term γ) if both x and y are in V (G)(resp. both x and y are in U(G), or one of x and y is in V (G) and the other is in U(G)). The blocktransformation-graph is defined on the vertex set " $ % . Two vertices x and y of are joined by an edge if and only if their associativity in G is consistent with the corresponding term of αβγ. Since there are eight distinct 3-permutation of {1, 0}, one can obtain eight blocktransformation graphs. It is interesting to see that is just the total-block graph of G, and . is the complement of . Each of these eight kinds of block-transformation graphs appears to have some nice properties; for instance, there diameters are small in most cases1. Structural properties, traversability and planarity of some of these blocktransformation graphs have been studied in2, 3, 4. In this paper, we establish a necessary and sufficient condition for the block-transformation graph to have crossing number 1. April, 2015 | Journal of Computer and Mathematical Sciences | www.compmath-journal.org.
(4) B. Basavanagoud, et al., J. Comp. & Math. Sci. Vol.6 (4), 222-227 (2015). 224. Definition 1.1. The block-transformation graph of a graph G is the graph with vertex set " $ % in which the vertices x and y are joined by an edge if one of the following conditions holds: &, ' ( " , and x and y are adjacent in G. (ii) &, ' ( % , and x and y are adjacent in G. (iii) One of x and y is in V (G) and the other is in U(G), and they are not incident in G, (see, Figure 1.1).. Figure 1.1. In , the light vertices correspond to the vertices G and dark vertices correspond to blocks of G. 2. CROSSING NUMBER 1 Theorem 2.A6 A graph is planar if and only if it has no subgraph homeomorphic to )* or )+,+ . Theorem 2.B4 Let G be a connected graph. The block-transformation graph is planar if and only if one of the following conditions holds: 1. G is a planar block. 2. G has exactly two blocks, each is outerplanar. 3. ! ,- , ) ,+ , . , where is a triangle whose any two vertices each of which is adjoined with an endedge and is a triangle whose each of the vertices is adjoined with an endedge. Theorem 2.1 The block-transformation graph of a connected nonplanar graph G with at least two blocks, has crossing number at least 3. April, 2015 | Journal of Computer and Mathematical Sciences | www.compmath-journal.org.
(5) 225. B. Basavanagoud, et al., J. Comp. & Math. Sci. Vol.6 (4), 222-227 (2015). Proof. Suppose G is a connected nonplanar graph with at least two blocks. Without loss of generality, let G has two blocks and with a common cutvertex v, such that is isomorphic to )* or )+,+ and ! ). By Kuratowski’s theorem on planar graphs, it is sufficient to prove that has crossing number at least 3. We see that in , the blockvertex is adjacent with all the incident vertices of block of G except v and the blockvertex is adjacent with the endvertex of block , also the block-vertex is adjacent with block-vertex forming subgraph homeomorphic to )/ or )-,+ , respectively, (see, Figure 1.2). Thus Cr( ) ≥ 3.. Theorem 2.2 The block-transformation graph of a connected graph G has crossing number 1 if and only if it satisfies one of the following conditions : 1. G is a block with crossing number 1. 2. G has 2 blocks such that one of the blocks is theta-minimally nonouterplanar and the other is outerplanar. 3. G has 3 blocks and is one of the following: (i) G is a triangle whose any one vertices is adjoined with two endedges. (ii) G is an outerplanar block of order 3 or 4 whose any two adjacent vertices each of which is adjoined to an endedge. (iii) G is a triangle whose one of the vertices is adjoined with a triangle and its another vertex is adjoined with an endedge. 4. G is a path ,- whose one of the nonendvertices is adjoined with an endedge. Proof. Suppose has crossing number 1. Let G has at least 4 blocks. Then has subgraphs homeomorphic to ,*, ) ,- , . , where has 2 cutvertices such that at least one of the blocks is not an edge and has 3 cutvertices incident with the same block. Then . we see that Cr(,* ) = 2, Cr() ,) = 5 and Cr( ) ≥ 2, a contradiction. Let is a triangle April, 2015 | Journal of Computer and Mathematical Sciences | www.compmath-journal.org.
(6) B. Basavanagoud, et al., J. Comp. & Math. Sci. Vol.6 (4), 222-227 (2015). 226. together with 3 vertices adjoined to the three vertices of the triangle. Then by Theorem 2.B, is planar. Let any of the nonendblocks in is not an edge, then we see that Cr( ) ≥ 2, a contradiction. Next we observe the following cases: Case 1. G is a block. Assume Cr(G) = k, k ≠ 1. Then = $ ) , hence Cr( ) = k ≠ 1, a contradiction. Case 2. G has two blocks, say and . If any of the blocks is nonplanar, then by Theorem 2.1, Cr( ) ≥ 3, a contradiction. If both are outerplanar then by Theorem2.B, G is planar, a contradiction. Let one of the blocks say is outerplanar and is not theta-minimally nonouterplanar. It is easy to verify that in the planar drawing of , the block-vertex forms two crossing when made to adjacent to the inner vertex, a contradiction. Case 3. G has 3 blocks. If any of the blocks is nonplanar, then by Theorem 2.1, Cr( ) ≥ 3, a contradiction. Assume one of the blocks is nonouterplanar. Then in , we have two block-vertices adjacent to the inner vertices giving at least 2 crossings. Therefore, G is outerplanar. Next we observe the following cases : Subcase 3.1. G has a single cutvertex. Suppose G = ) ,+ . Then by Theorem 2.B, is planar a contradiction. Let one of the blocks is a cycle 01 , n ≥ 4, and the other two blocks are edges. Then it is easy to see that has 2 crossings. Let at least two of the blocks are cycles. Without loss of generality, assume G has two triangles and an edge. Then it is easy to see that has 2 crossings, a contradiction. Subcase 3.2. G has 2 cutvertices. Suppose G = ,-. By Theorem 2.B, is planar, a contradiction. Let the two endblocks be cycles, then we see that Cr( ) ≥ 2. Let G is a cycle 01 , n ≥ 4, then it is easy to check that Cr( ) ≥ 2. Let the nonendblock is of order ≥ 5. Then also Cr( ) ≥ 2. Let one of the endblocks and the nonendblock are cycles such that at least on of them is of order ≥ 4. We again see that Cr( ) ≥ 2. In all these cases we arrive at a contradiction. Conversely, let G be a block with crossing number 1. Then = $ ) which has crossing number 1. Let G has 2 blocks, say and such that is theta-minimally nonouterplanar. Then in , the block-vertex is adjacent to all the vertices of block except v giving a subgraph homeomorphic to )+,+ or )*. Also, we observe that the blockvertex is adjacent with the vertices of block forming a outerplanar subgraph and blockvertices and are adjacent, hence has crossing number 1. Let G satisfies condition 3(i), (ii) or (iii). Then has a subgraph homeomorphic to )* formed by the vertices a, b, c, d, e, giving only one crossing, (see, Figure 1.3 (a), (b), (c) respectively). Suppose G is a path ,- with a vertex adjoined to a nonendvertex of the path. Then has a subgraph homeomorphic to )* formed by the vertices a, b, c, d, e, giving only one crossing, (see, Figure 1.3 (d)). April, 2015 | Journal of Computer and Mathematical Sciences | www.compmath-journal.org.
(7) 227. B. Basavanagoud, et al., J. Comp. & Math. Sci. Vol.6 (4), 222-227 (2015). Figure 1.3. ACKNOWLEDGEMENT This research was supported by UGC-MRP. F.No.41-784/2012(SR) dated 17-072012. REFERENCES 1. B. Basavanagoud, H. P. Patil, Jaishri B. Veeragoudar, On the blocktransformation graphs, graph-equations and diameters, International Journal of Advances in Science and Technology, 2(2), 62-74 (2011). 2. B. Basavanagoud, H. P. Patil, Jaishri B. Veeragoudar, Traversability and Planarity of the Block-Transformation Graph when αβγ = 101, International Transactions on Electrical, Electronics and Communication Engineering, 1(6), 49-56 (2011). 3. B. Basavanagoud, H. P. Patil, Jaishri B. Veeragoudar, Blocktransformation graph when αβγ = 100, Indian Journal of Mathematics, 55(3), 297-313 (2013). 4. B. Basavanagoud, H. P. Patil, Jaishri B. Veeragoudar, Blocktransformation graph when αβγ = 110, Journal of Discrete Mathematical Sciences and Cryptography, 17(1), 55-68 (2014). 5. R. K. Guy, Latest results on crossing numbers in recent trends in graph theory, Springer, New York, 143-156 (1971). 6. F. Harary, Graph theory, Addison-Wesley, Reading, Mass, (1969). 7. V. R. Kulli, The semitotal-block graph and the total-block graph of a graph, Indian J. Pure. Appl. Math, 7, 625-630 (1976). 8. V. R. Kulli, On minimally nonouterplanar graphs, Proc. Indian. Nat. Sci. Acad, 41A, 275-280 (1975).. April, 2015 | Journal of Computer and Mathematical Sciences | www.compmath-journal.org.
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