Qlick Graphs with Crossing Number One

In this paper, we deduce a necessary and sufficient condition for graphs whose qlick graphs have crossing number one. We also obtain a necessary and sufficient condition for qlick graphs to have crossing number one in terms of forbidden subgraphs.


Introduction
A graph is planar if it can be drawn in the plane or on the sphere in such a way that no two of its edges intersect.The crossing number ) (G cr of a graph G is the least number of intersections of pairs of edges in any embedding of G in the plane.Obviously, G is planar if and only if . It is implicit that the edges in a drawing are Jordan arcs (hence, non-selfintersecting), and it is easy to see that a drawing with the minimum number of crossing (an optimal drawing) must be a good drawing, that is, each two edges have at most one point in common, which is either a common end-vertex or a crossing.
All graphs considered here are finite undirected and without loops or multiple edges.We refer to [6] for unexplained terminology and notations.For a graph G , let e and B and so on.If two distinct blocks 1 B and 2 B are incident with common cut-vertex, then they are adjacent blocks.The edges and blocks of a graph are called its members.
The qlick graph ) (G Q of a graph G is the graph whose set of vertices is the union of the set of edges ) (G

E
and set of blocks ) (G U of G and in which two vertices are adjacent if and only if the corresponding edges are adjacent or corresponding blocks are adjacent or an edge and a block are incident.The plick graph P(G) of G is the graph whose vertex set is the set of edges ) (G

E
and set of blocks ) (G U of G and two vertices are adjacent if and only if the corresponding edges are adjacent or an edge and a block are incident.These concepts were introduced by Kulli in [10] and were studied, for example, in [1,2,11,12]. In [8], Kulli introduced the concept of total-block graph of a graph.In [3], Basavanagoud et al. generalized the concept of total-block graph and introduced the block-transformation graphs and defined as follows: Let  = (, ) be a graph with block set (), and let , ,  be three variables having values 0 or 1.The block-transformation graph   is the graph having () ∪ () as the vertex set.For any two vertices  and  ∈ () ∪ () we define , ,  as follows: Thus, we obtain eight kinds of block -transformation graphs, in which  111 is the total-block graph of  [8], and  000 is its complement.Also,  001 ,  010 and  011 are the complements of  110 ,  101 and  100 respectively.In [4], Basavanagoud et al. studied a criterian for (non-) planarity of the block-transformation graph   when  = 101.
The line graph of a graph G is the graph whose vertices can be put in one to one correspondence with the edges of G in such a way that two vertices of are adjacent if and only if the corresponding edges of G are adjacent.The block graph of a graph G is the graph whose vertices are the blocks of G and in which two vertices are adjacent whenever corresponding blocks have a cut-vertex in common.For other definitions concerning crossing numbers, line graphs or block graphs see [6].
The following will be useful in the proof of our results.
Remark 1.1 [11].For any graph G , Theorem 1.1 [13].The line graph of a planar graph G is planar if and only if and every vertex of degree 4 is a cut vertex.Theorem 1.2 [7].Let G be a non planar graph.Then if and only if the following conditions hold: , and every vertex of degree 4 is a cut-vertex of G ; (3) There exists a drawing of G in the plane with exactly one crossing in which each crossed edge is incident with a vertex of degree 2. Theorem 1.3 [9].The line graph of a planar graph G has crossing number one if and only if (1) or (2)  holds: and there is unique non-cut-vertex of degree 4; , every vertex of degree 4 is a cut-vertex, there is a unique vertex of degree 5 and it has at most 3 incident edges in any block.Theorem 1.4 [11].The qlick graph ) (G Q of a graph G is planar if and only if G satisfies the following conditions: , and

Results
The following theorem supports the main theorem.

Theorem 2.1 Let x be any edge of
G is the graph as in the statement.Referring to Figure 1, it is immediate to see that 1 = )) ( ( The following theorem gives a characterization of qlick graphs with crossing number 1.

Theorem 2.2 A graph
G has a qlick graph with crossing number 1 if and only if G is planar and one of the following holds: , G has exactly two adjacent non-cut-vertices of degree 3 and every other vertex of degree 3 is a cut-vertex. (2 , every vertex of degree 3 or 4 is a cut-vertex and there is a unique vertex v of degree 4 and v lies on 2 or 3 blocks., then there exists a non-cut-vertex of degree 3 or 4 and more than one vertex of degree 4 and v lies on 2 or 3 blocks.Suppose We now consider the following cases: . Then by Theorem 1.4 and since , G has a non-cut-vertex of degree 3. Clearly G contains a subgraph homeomorphic to x K  4 , so that there exist at least two non-cut-vertex degree 3 .More precisely there is an even number, say n 2 , of non-cut-vertex degree 3 .Now suppose G has at least two diagonal edges.Then there are two subcases to consider depending on whether 2 diagonal edges exist in one cycle or in two different edge disjoint cycles.has at least 2 crossings, a contradiction.Hence G has exactly two non-cut-vertices of degree 3 and every other vertex of degree 3 is a cut-vertex.
Suppose G has two non-cut-vertices of degree 3 and they are not adjacent.Then G contains a subgraph homeomorphic to 2,3 has crossing number exceeding 1, a contradiction.(see Figure 2) Bulletin of Mathematical Sciences and Applications Vol. 17 Therefore, we conclude that G contains exactly two non-cut-vertices of degree 3 and these are adjacent.This proves(1).

Case 2. Suppose
. Then every vertex of degree 3 is a cut-vertex, otherwise, by case 1, Q(G) has crossing number exceeding 1.
Next we show that every vertex of degree 4 is a cut-vertex.On the contrary suppose that G has non-cut-vertex v of degree 4 .Then by Theorem . The vertex 1 u in ) (G Q corresponding to the block which contains a non-cut-vertex of degree 4 is adjacent to every vertex of ) (G L . We obtain the drawing of ) (G Q with 3 crossings.Assume v lies on at least 4 blocks.The vertex v and four of its neighboring vertices lie in a subgraph of has crossing number exceeding 1, a contradiction.
Suppose G has at least two cut-vertices 1 u and 2 u which are of degree 4 .Then the cut-vertices 1 u and 2 u together with their corresponding incident four edges form two subgraphs as G is a planar graph satisfying (1) or (2) .Then by Theorem 1.4, ) (G Q has crossing number at least 1.We now show that its crossing number is at most 1.First assume (1)  holds.Then G has exactly one subgraph 1 H , homeomorphic to x K  4 which contains exactly two adjacent non-cut-vertices of degree 3 .By Theorem 2. has crossing number 1. Now assume (2) holds, and let v be a unique cut-vertex of degree 4 and it lies on 2 or 3 blocks.We have the following cases.
Case 1. Suppose v lies on 2 blocks.Then each block of G is a cycle.On embedding of has crossing number 1.

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Case 2. Suppose v lies on 3 blocks.Then one of these blocks is a cycle and each of the remaining blocks is a 2 K .The qlick graph ) (G Q can be constructed in the plane which gives exactly one subgraph 5 K (see Figure 5).Hence

Forbidden Subgraphs
By using Theorem 2.2, we now characterize graphs whose qlick graphs have crossing number one in terms of forbidden subgraphs.

Theorem 3.1 The qlick graph
) (G Q of a connected graph G has crossing number one if and only if G has no induced subgraph 1,4 K or a subgraph homeomorphic from any one of the graphs Figure 6.

Bulletin of Mathematical Sciences and Applications Vol. 17
Proof.Suppose G is a connected plane graph such that ) (G Q has crossing number one.By Theorem 2.2, we have (1) 3 = ) (G  , G has exactly two adjacent non-cut-vertices of degree 3 and every other vertex of degree 3 is a cut-vertex or (2) , every vertex of degree 3 or 4 is a cut-vertex and there is unique vertex v of degree 4 lies on 2 or 3 blocks.From (1) and (2) it follows that G has no induced subgraph 1,4 K or a subgraph homeomorphic from any one of the graphs of Fig 6 .Conversely, suppose G is a connected plane graph and it does not contain 1,4 K as induced subgraph or a subgraph homeomorphic from any one of the graphs of Fig 6 .We now show that G satisfies the condition (1) or (2) and hence by Theorem 2.1, ) (G Q has crossing number one.Suppose there exist at least three non-cut-vertices of degree 3 in G .Then G has at least two diagonal edges.Here we consider two cases depending on whether 2 diagonal edges exist in one block or in two different blocks.In each case we arrived at a contradiction.Hence G has exactly two non-cut-vertices of degree 3 .
Suppose G has exactly two nonadjacent non-cut-vertices of degree 3 .Then there exist 3 disjoint paths between these two non-cut-vertices of degree 3 .Clearly G contains a subgraph homeomorphic from 4 G , a contradiction.Thus G has exactly two adjacent non-cut-vertices of degree 3 . .Then G contains a subgraph homeomorphic from 5 G , a contradiction.
Let v be a vertex of G and deg 4 = v . We prove that v is a cut-vertex.On the contrary, let c b a , , and d be the vertices of G adjacent to v .Then there exist paths between every pair of vertices of c b a , , and d not containing v .Then it is proved in Theorem 1.5, G has a subgraph homeomorphic from G as a subgraph and G as a subgraph, this is a contradiction.Thus v is a cut-vertex and every vertex of degree 4 is a cut-vertex.
Suppose every cut-vertex of degree 4 lies on 4 blocks of G .Let v be the cut-vertex of degree 4 and it lies on 4 blocks.Then 1,4 K is an induced subgraph of G , a contradiction.Hence v lies on 2 or 3 blocks of G .
Suppose there are at least two cut-vertices of degree 4 , each which lies 2 or 3 blocks.Let 1 v and 2 v be the two cut-vertices of degree 4 such that 1 v and 2 v are connected by a path P and let i a , 1,2,3 = i and j b , 1,2,3 = j be the vertices of adjacent to 1 v and 2 v respectively.
We consider the following cases.

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Case 1. Assume 1 v and 2 v both lies on 3 blocks.If there exists a path between a vertex of i a and a vertex of j b , then G has a subgraph homeomorphic from 6 G .Case 2. Assume 1 v and 2 v both lies on 2 blocks.If there exists a path between a vertex of i a and a vertex of j b and there is a path between remaining two vertices of G as a subgraph.We have exhausted all the cases and we arrive at the conclusion that G has exactly one cut-vertex of degree ].
We have completed all cases.In each case, we found that G contains a subgraph homeomorphic from one of the forbidden subgraphs of Fig. 6 .Hence 1 v is a cut-vertex.Thus Theorem 2.2 implies that G has a qlick graph with crossing number one.

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In this paper two characterizations are given for graphs whose qlick graphs have crossing number 1, one in terms of degrees of vertices of a graph and the second in terms of forbidden subgraph.The crossing number cr(G) of G is the least number of intersections of pairs of edges in any embedding of G in the plane.Obviously G is planar if and only if cr(G)=0.

Case 1 .Case 2 .
If two diagonal edges exist in one block of G , then G has a subgraph homeomorphic from 1 G or 2 G .If two diagonal edges exist in two different blocks of G , then G has a subgraph homeomorphic from 3 G .
u and a block B are incident with each other, as are 2 u and B and so on.If B ={ 1  s} is a block of G , then we say that edge 1 e and block B are incident with each other, as are 2 Suppose ,  are in ().=1 if  and  are adjacent in .=0 if  and  are nonadjacent in .(ii) Suppose ,  are in ().=1 if  and  are adjacent in .=0 if  and  are nonadjacent in .()  ∈ () and  ∈ ().=1 if  and  are incident with each other in .=0 if  and  are nonincident with each other in .

Bulletin of Mathematical Sciences and Applications Vol. 17 Subcase 2.3.1.
4which lies on either 2 or 3 blocks of G .Assume 2 Z and 3Z are not disjoint.Let x be the last but one vertex of 2 Z which also belongs to Z. Again we consider two subcases of subcase 2.3.v lies on three blocks.] I