ON PLANARITY OF 3-JUMP GRAPHS

For a graph G of size m ≥ 1 and edge-induced subgraphs F and H of size k where 1 ≤ k ≤ m, the subgraph H is said to be obtained from the subgraph F by an edge jump if there exist four distinct vertices u, v, w and x such that uv ∈ E(F ), wx ∈ E(G)−E(F ), and H = F − uv + wx. The k-jump graph Jk(G) is that graph whose vertices correspond to the edge-induced subgraphs of size k of G where two vertices F and H of Jk(G) are adjacent if and only if H can be obtained from F by an edge jump. All connected graphs G for whose J3(G) is planar are determined. AMS Subject Classification: 05C10, 05C12


Introduction
The concept of the k-jump graph of a nonempty graph G of size m where 1 ≤ k ≤ m was introduced by Chartrand, Hevia, Jarrett and Schultz [1].Let G be a graph of size m ≥ 1 and F and H be edge-induced subgraphs of size k of G where 1 ≤ k ≤ m.The subgraph H is said to be obtained from the subgraph F by an edge jump if there exist four distinct vertices u, v, w and x such that uv ∈ E(F ), wx ∈ E(G) − E(F ), and H = F − uv + wx.It is obvious that if H is obtained from F by an edge jump then F is obtained from H by an edge jump.If there is a sequence F = H 0 , H 1 , . . ., H ℓ = H where ℓ ≥ 1 of edge-induced subgraphs of size k such that H i+1 is obtained from H i by an edge jump for 0 ≤ i ≤ ℓ − 1, then we say that F can be j-transformed into H.The minimum number of edge jumps required to j-transform F into H is the k-jump distance from F to H.For a graph G of size m ≥ 1 and an integer k with 1 ≤ k ≤ m, the k-jump graph J k (G) is that graph whose vertices correspond to the edge-induced subgraphs of size k of G where two vertices F and H of J k (G) are adjacent if and only if the k-jump distance between edge-induced subgraphs F and H is 1 that is, H is obtained from F by an edge jump.We can label each vertex of J k (G) by listing all the edges of the respective subgraph.The concept of the k-jump graph is illustrated in Figure 1.In particular, if k = 1 then the graph J 1 (G) = J(G) is called the jump graph of G.Moreover, J(G) = L(G), the complement of the line graph of G.In [3] all connected graphs G for whose J 2 (G) is planar are determined in terms of a finite set S of graphs, namely a connected graph G has a planar 2-jump graph if and only if G is a subgraph of some element of S. The goal of this paper is to characterize all connected graphs having a planar 3-jump graph along the same lines as the characterization of connected graphs having a planar 2-jump graph.The following results appeared in [3] and [4] will be useful for us later.
Theorem 1.1.( [4]) A graph is planar if and only if it contains no subgraph isomorphic to K 5 or K 3,3 or a subdivision of one of these graphs.
The reader is referred to the book [2] by Chartrand, Lesniak and Zhang for basic definitions and terminology not described here.

Connected Graphs with Planar 3-Jump Graphs
In this section we will focus our attention on the planarity of the 3-jump graph J 3 (G) for a connected graph G of size at least 3.As we mentioned earlier, our aim is to determine a finite set S of graphs with the property that a connected graph G has a planar 3-jump graph if and only if G is a subgraph of some element of S.
In [3] it is shown that for the path P n and cycle C n of order n, J 2 (P n ) is nonplanar if and only if n ≥ 6 and J 2 (C n ) is nonplanar if and only if n ≥ 5. Similar results can be obtained for J 3 (P n ) and J 3 (C n ).The 3-jump graphs J 3 (P 5 ), J 3 (P 6 ), J 3 (C 4 ), and J 3 (C 5 ) are shown in Figure 2 which we see that J 3 (P 5 ) and J 3 (C 4 ) are planar while J 3 (P 6 ) and J 3 (C 5 ) are nonplanar.Therefore, by Theorems 1.1 and 1.3, the following results are immediate.
We now present a simple but useful lemma.Proof.Since there exists a subgraph of the 3-jump graph J 3 (H) that is isomorphic to a subdivision of K 3,3 as shown in Figure 3, it follows that J 3 (H) and so J 3 (G) are nonplanar.Proof.Since the 3-jump graph J 3 (H) contains a subgraph isomorphic to a subdivision of K 3,3 as shown in Figure 4, it follows that J 3 (H) and thus J 3 (G) are nonplanar.Proof.We have seen in Corollaries 2.1 and 2.2 that J 3 (P 6 ) and J 3 (C 5 ) are nonplanar.Thus it remains to show the nonplanarity for J 3 (N i ) where 1 ≤ i ≤ 18.
For i ∈ {1, 5, 11, 16, 18}, some subgraph of J 3 (N i ) is shown in Figure 8(a), (b), (c), (d) and (e), respectively.Since, for each i, J 3 (N i ) contains a subgraph that is isomorphic to either a subdivision of K 5 or a subdivision of K 3,3 , J 3 (N i ) is nonplanar.
For i ∈ {2, 3, 4, 7, 8, 12, 13, 15, 17}, N i contains subgraphs G 1 , G 2 and G 3 as mentioned in Lemma 2.6.(all edges of N i are labeled to be corresponding with the edges of G 1 , G 2 and G 3 in Lemma 2.6.)Thus, by Lemma 2.6, it follows that J 3 (N i ) is nonplanar.
For i ∈ {6, 9}, N i contains a subgraph H as mentioned in Lemma 2.4.(all edges of N i are labeled to be corresponding with the edges of H 1 and H 2 in Lemma 2.4.)Thus, by Lemma 2.4, J 3 (N i ) is nonplanar.
For i ∈ {10, 14}, N i contains a subgraph H as mentioned in Lemma 2.5.(all edges of N i are labeled to be corresponding with the edges of H 1 and H 2 in Lemma 2.5.)Thus, by Lemma 2.5, J 3 (N i ) is nonplanar.
Proof.If H = G then the result is trivial.Assume that H is a proper subgraph of G and so m−m ′ ≥ 1.We show that J k+m−m ′ (G) contains a subgraph F isomorphic to J k (H) which is nonplanar.Let e 1 , e 2 , . . ., e m−m ′ ∈ E(G) − E(H).Now, for each vertex X of J k (H), let e 1 e 2 . . .e m−m ′ X be a vertex of F .Thus V (F ) ⊆ V (J k+m−m ′ (G)).Since e 1 e 2 . . .e m−m ′ X and e 1 e 2 . . .e m−m ′ Y are adjacent in F if and only if X and Y are adjacent in J k (H), it follows that F is isomorphic to J k (H).
In [3] it has been shown that for each graph H i where 1 ≤ i ≤ 17 in Figure 9 We now turn our attention to graphs having a planar 3-jump graph.If Thus we have an immediate result.
Proposition 2.9.If G is a star of size m then J k (G) is planar for every k where 1 ≤ k ≤ m.Proof.The 3-jump graph J 3 (M i ) of a graph M i is shown in Figure 11 for 1 ≤ i ≤ 5, and in Figure 12 for 6 ≤ i ≤ 11.Thus, for each i, J 3 (M i ) is planar.
Next, we investigate that these graphs M i , where 1 ≤ i ≤ 11, of Figure 10 are maximal in the sense that its 3-jump graph is planar.
For the converse, let G be a connected graph that is not a star for whose J 3 (G) is planar.Then the connected graph G may or may not contain cycles.
Case 1. G is a tree.Since G cannot contain N 20 = P 6 , it follows that diam(G) ≤ 4 and since G is not a star, we have that diam(G) ≥ 3. Thus either diam(G) = 3 or diam(G) = 4.If diam(G) = 3 then G is a double star.Observe  We next present another characterization of graphs for whose the 3-jump graph is planar and also show that these graphs N i , for 1 ≤ i ≤ 20 are minimal according to its 3-jump graph being nonplanar.
Corollary 2.12.For a connected graph G that is not a star, the 3-jump graph J 3 (G) is planar if and only if G does not contain any of N i for 1 ≤ i ≤ 20 of Figure 7 as a subgraph.
Proof.If G contains N i for some i where 1 ≤ i ≤ 20, then J 3 (G) contains J 3 (N i ), by Lemma 2.3, and so J 3 (G) is nonplanar since J 3 (N i ) is nonplanar.
For the converse, we assume that G does not contain any of N i for 1 ≤ i ≤ 20.We consider two cases.
Case 1. G is a tree.Since G does not contain N 20 = P 6 , it follows that diam(G) ≤ 4 and since G is not a star, we have that diam(G) ≥ 3. Thus either Next, if G contains both C 3 and C 4 then since none of N i for i ∈ {10, 11, 19, 20} is contained in G, G is a subgraph of M 6 or M 7 and thus J 3 (G) is planar.Finally, if G contains C 3 but not C 4 then since G does not contain N i where 12 ≤ i ≤ 18 and N 20 , G is a subgraph of M i for some 8 ≤ i ≤ 11 and thus J 3 (G) is planar.

Final Remarks
In this paper, we have characterized connected graphs whose the 3-jump graph is planar.A natural question arises what the characterization of a connected graph whose the k-jump graph where 4 ≤ k ≤ m − 4 is planar.

Figure 1 :
Figure 1: The k-jump graphs of a graph

Theorem 1 .
1 and lemma 2.3 give us the following results.Lemma 2.4.If G is a graph containing a subgraph H where H is the union of edge-disjoint subgraphs H 1 of size at least 2 and H 2 of size at least 4 such that (1) H 1 contains edges b and e, (2) H 2 contains edges a, c, d and f where a is not adjacent to c, and (3) edges b and e are not adjacent to both d and f in H, then J 3 (G) is nonplanar.

Figure 5 :
Figure 5: The four possible subgraphs for a graph G of Lemma 2.6

Figure 11 :
Figure 11: The 3-jump graphs of M i for 1 ≤ i ≤ 5

Figure 15 :
Figure 15: All graphs that contain C 4 but not C 3 for whose the 3-jump graph is planar

Figure 17 :
Figure 17: All graphs that contain C 3 but not C 4 for whose the 3-jump graph is planar