Journal of Computer and Mathematical Sciences, Vol.6(7),388-394, July 2015 ISSN 0976-5727 (Print) (An International Research Journal), www.compmath-journal.org ISSN 2319-8133 (Online)

On Semifull Line Graphs and Semifull Block Graphs

V. R. Kulli

Department of Mathematics, Gulbarga University, Gulbarga, INDIA. email: [email protected]

(Received on: July 17, 2015)

ABSTRACT

In this paper, we introduce the concepts of (i) the semifull line graph of a graph and (ii) the semifull block graph of a graph. We obtain some properties of these graphs. We obtain characterizations of graphs (i) whose semifull line graphs and semifull graphs are isomorphic (ii) whose semifull block graphs and semifull graphs are isomorphic.

Mathematics Subject Classification : 05C

Keywords: middle line graph, middle block graph, semifull line graph, semifull block graph, semifull graph.

1. INTRODUCTION

All graphs considered here are finite , undirected without loops or multiple lines. Any undefined term or notation in this paper may be found in Kulli 1.

If B = {u1, u2, …, ur, r ≥ 2} is a block of a graph G, then we say that point u1 and block B are incident with each other, as are u2 and B and so on. If two blocks B1 and B2 of G are incident with a common cutpoint, then they are adjacent blocks. If B = { e1, e2, ... es, s≥1} is a block of a graph G, then we say that line e1 and block B are incident with each other, as 2 are e2 and B and so on. This idea was introduced by Kulli in . The points, lines and blocks of a graph are called its members. The middle line graph Ml(G) of a graph G is the graph whose point set is the union of the set of points, lines and blocks of G in which two points are adjacent if the corresponding lines are adjacent or one corresponds to a point and the other to a line incident with it or one corresponds to a block B of G and the other to a point v of G and v is in B. This concept was introduced by Kulli in 3 Many other graph valued functions in graph theory were studied, for example, in 4,5,6,7,8,9,10,11,12,13,15,16,17,18 and also graph valued functions in domination theory were studied, for example, in 19,20,21,22,23,24,25,26,27 .

June, 2015 | Journal of Computer and Mathematical Sciences | www.compmath-journal.org 389 V. R. Kulli, J. Comp. & Math. Sci. Vol.6(7), 388-394 (2015)

The middle block graph Mb(G) of a graph G is the graph whose point set is the union of the set of points, lines and blocks of G in which two points are adjacent if the corresponding blocks contain a common cutpoint of G or one corresponds to a block B of G and the other to a point v of G and v is in B or one corresponds to a point and the other to a line incident with it. This concept was introduced by Kulli in 3. The middle blict graph Mn(G) of a graph G is the graph whose point set is the union of the set of points, lines and blocks of G in which two points are adjacent if the corresponding lines of G are adjacent or the corresponding blocks of G are adjacent or one corresponds to a point and other to a line incident with it or one corresponds to a block B of G and other to a point v of G and v is in B. This concept was introduced by Kulli and Biradar in 27. The block line forest Bf(G) of a connected graph G is the graph whose point set is the union of the set of lines and the set of blocks of G in which two points are adjacent if one corresponds to a block of G and other to a line incident with it. This concept was introduced by Kulli in 8. The block graph B(G) of a graph G is the graph whose point set is the set of blocks of G in which two points are adjacent if the corresponding blocks are adjacent. This graph was studied, for example, in 29,30,31 . The line graph L(G) of a graph G is the graph whose point set is the set of lines of G in which two points are adjacent if the corresponding lines are adjacent. This graph was studied, for example, in 32 33,34,35 . The semifull graph Fs(G) of a graph G is the graph whose point set is the union of points, lines and blocks of G in which two points are adjacent if the corresponding members of G are adjacent or one corresponds to a point and the other to a line incident with it or one corresponds to a block B of G and the other to a point v of G and v is in B. This concept was introduced in 36. The following will be useful in the proof of our results.

36 Theorem A . If G is a connected ( p, q) graph whose points have degree di and if bi is the number of blocks to which point vi belongs in G, then the semifull graph Fs(G) of G has 1 1 q+∑ b + 1 points and 2q+∑ b2 + ∑ bb() + 1 lines. i 2i 2 i i

3 Theorem B . If G is a ( p, q) graph whose points have degree di and bi is the number of + + blocks to which point vi belongs in G, then the middle line graph Ml(G) of G has q∑ b i 1 1 points and q+∑ b + ∑ d 2 lines. i2 i

3 Theorem C . If G is a ( p, q) graph whose points have degree di and bi is the number of + + blocks to which point vi belongs in G, the middle block graph Mb(G) of G has q∑ b i 1 1 points and 2q+∑ b( b + 1 ) lines. 2 i i Theorem D 1. A nontrivial graph is bipartite if and only if all its cycles are even.

July, 2015 | Journal of Computer and Mathematical Sciences | www.compmath-journal.org V. R. Kulli, J. Comp. & Math. Sci. Vol.6(7), 388-394 (2015) 390 2. SEMIFULL LINE GRAPHS

We now define the following graph valued function.

Definition 1. The semifull line graph Fl(G) of a graph G is the graph whose point set is the union of the set of points, lines and blocks of G in which two points are adjacent if the corresponding points and lines of G are adjacent or one corresponds to a point of G and other to a line incident with it or one corresponds to a block B of G and other to a point v of G and v is in B. Example 2. In Figure 1, a graph G and its semifull line graph Fl(G) are shown.

e1 e2

v e1 e 2 2 v v1 3 v1 B1 v2 B2 v3

B B 1 2 Figure 1

Remark 3. If G is a connected graph, then Fl(G) is also connected and conversely.

Remark 4. The semifull line graph Fl(G) of G is a spanning subgraph of the semifull graph Fs(G) of G.

Remark 5. For any graph G, Fl(G) = Bl(G) ∪ L(G) ∪ G.

Theorem 6. For any graph G, Fl(G) = Ml(G) ∪ G.

Theorem 7. For any graph G, Fs(G) = Fl(G) ∪ B(G). We now establish a result which determines the number of points and lines in Fl(G).

Theorem 8. If G is a ( p, q) connected graph, whose points have degree di and if bi is the number of blocks to which point vi belongs in G, then the semifull line graph Fl(G) of G has 1 q+∑ b + 1 points and 2q+∑ b + ∑ d 2 lines. i i2 i

Proof: By Remark 4, Fl(G) is a spanning subgraph of Fs(G). Thus the number of points of + + Fl(G) equals the number of points of Fs(G). By Theorem A, Fs(G) has q∑ b i 1 points.

Hence the number of points in Fl(G) + + = q∑ b i 1.

By Theorem 6, the number of lines in Fl(G) is the sum of the number of lines in

1 2 Ml(G) and the number of lines in G. By Theorem B, Ml(G) has q+∑ b + ∑ d lines. Also i2 i G has q lines. Thus the number of lines in Fl(G) 1 = 2q+∑ b + ∑ d 2 . i2 i

July, 2015 | Journal of Computer and Mathematical Sciences | www.compmath-journal.org 391 V. R. Kulli, J. Comp. & Math. Sci. Vol.6(7), 388-394 (2015) We characterize graphs whose semifull line graphs and semifull graphs are isomorphic.

Theorem 9 . Let G be a nontrivial connected graph. The graphs Fl(G) and Fs(G) are isomorphic if and only if G is a block.

Proof: Let G be a ( p, q ) connected graph and be a block. The graphs Fl(G) and Fs(G) have the same number of points. Since G is a block and B(G) has no lines, it implies, by definitions, that Fl(G) and Fs(G) are isomorphic. Conversely suppose Fl(G) = Fs(G) and G is a nontrivial connected graph. We now prove that G is a block. On the contrary, assume G has at least two blocks. By Theorem 7, it is clear that the number of lines in Fs(G) is the sum of the number of lines in Fl(G) and the number of lines in B(G). Since G has at least two blocks, It implies that B(G) has at least one line. Hence the number of lines in Fl(G) is less than the number of lines in Fs(G). Thus Fl(G)

≠ Fs(G), which is a contradiction. Thus G has no two or more blocks and hence G is a block.

Corollary 10. Let G be a graph without isolated points. The graph Fl(G) and Fs(G) are isomorphic if and only if every component of G is a block.

Theorem 11. If G is a graph without isolated points, then Fl(G) is not a bipartite graph.

Proof: If G is a graph without isolated points, then G has a line, say e = uv . Since e is incident with the points u and v, it implies that the corresponding points of e, u, v form a cycle C3 in Fl(G). By Theorem D, Fl(G) is not bipartite.

3. SEMIFULL BLOCK GRAPHS

We now define another graph valued function.

Definition 12. The semifull block graph Fb(G) of a graph G is the graph whose point set is the union of the set of points, lines and blocks of G in which two points are adjacent if the corresponding points and blocks of G are adjacent or one corresponds to a point of G and other to a line incident with it or one corresponds to a block B of G and other to a point v of G and v is in B.

Example 13. In Figure 2, a graph G and its semifull block graph Fb(G) are shown.

e1 e2

e1 e 2 v v1 3 v1 v B1 v2 B2 v3 2

B B 1 2 Figure 2

July, 2015 | Journal of Computer and Mathematical Sciences | www.compmath-journal.org V. R. Kulli, J. Comp. & Math. Sci. Vol.6(7), 388-394 (2015) 392

Remark 14. If G is a connected graph, then Fb(G) is also connected and conversely.

Remark 15. The semifull block Fb(G) is a spanning subgraph of Fs(G).

Theorem 16. For any graph G, Fb(G) = Mb(G) ∪ G.

Theorem 17. For any graph G, Fs(G) = Fb(G) ∪ L(G).

We now determine the number of points and lines in Fb(G).

Theorem 18. If G is a ( p, q) connected graph whose points have degree di and if bi is the number of blocks to which points vi belongs in G, then the semifull block graph Fb(G) of G 1 has q+∑ b + 1 points and 3q+∑ b() b + 1 lines. i 2 i i Proof: By Remark 15, the semifull block graph Fb(G) is a spanning subgraph of Fs(G). Thus the number of points of Fb(G) equals the number of points of Fs(G). By Theorem A, Fs(G) + + has q∑ b i 1 points. Hence the number of points in Fb(G) + + = q∑ b i 1.

By Theorem 16, the number of lines in Fb(G) is the sum of the number of lines in 1 Mb(G) and the number of lines in G. By Theorem C, Mb(G) has 2q+∑ b( b + 1 ) lines. Also 2 i i G has q lines . Hence the number of lines in Fb(G) 1 = 3q+∑ b() b + 1 . 2 i i We characterize graphs whose semifull block graphs and semifull graphs are isomorphic.

Theorem 19, Let G be a nontrivial connected graph. The graphs Fb(G) and Fs(G) are isomorphic if and only if G is P2.

Proof: Suppose G = P2. Then clearly Fb(G) = Fs(G) = K4 – x. Conversely suppose G is a nontrivial connected graph and Fb(G) = Fs(G). We now prove that G = P2. On the contrary, assume G has at least two lines. By Theorem 17, it is clear that the number of lines in Fs(G) is the sum of the number of lines in Fb(G) and the number of lines in L(G). Since G has at least two lines, it implies that L(G) has at least one line. Thus the number of lines in Fb(G) is less than the number of lines in Fs(G). Hence Fl(G) and Fs(G) are not Isomorphic, which is a contradiction. Thus G has no two are more lines and hence G is P2.

Corollary 20. Let G be a graph without isolated points. Then Fb(G) = Fs(G) if and only if G

= mP 2, m ≥ 1.

Theorem 21. If G is a graph without isolated points, then Fb(G) is not a bipartite graph.

July, 2015 | Journal of Computer and Mathematical Sciences | www.compmath-journal.org 393 V. R. Kulli, J. Comp. & Math. Sci. Vol.6(7), 388-394 (2015)

Proof: Let G be a graph without isolated points. Then G has a block B. Let points u, v ∈ B. Since B is incident with the points u and v, it implies that the corresponding points of B, u, v, form a cycle C3 in Fb(G). By Theorem D, Fb(G) is not a bipartite graph.

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