The Point Block Graph of A Graph V. R. Kulli and M. S. Biradar

Abbr:J.Comp.&Math.Sci.

2014, Vol.5(5): Pg.476-481

An International Open Free Access, Peer Reviewed Research Journal www.compmath-journal.org

The Point Block Graph of A Graph

V. R. Kulli and M. S. Biradar

1 Department of Mathematics, Gulbarga University, Gulbarga, INDIA.

2 Department of Mathematics,

Govt. First Grade College, Basavakalyan, INDIA.

(Received on: October 15, 2014)

ABSTRACT

In this paper, we introduce the concept of the point-block graph of a graph. We obtain some properties and a characterization of the point-block graph of graph. We present a characterization of those graphs whose point-block graphs and middle graphs are isomorphic. We establish some relationships between (i) point-block graph and block graph (ii) point-block graph and line graph and (iii) point-block graph and block-point tree.

Keywords: point-block graph, line graph, middle graph, block graph, block-point tree. AMS: Subject Classification: 05C10.

1. INTRODUCTION

In this paper, we consider a graph a finite, undirected without loops or multiple lines. We use the terminology of ² .

The point-block graph ^Pb ( ) ^{G of a}

graph G is the graph whose point set is the set of points and blocks of G and 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 . This} concept was introduce by Kulli and Biradar

in ¹² and was studied in ^14,15 . Many other graph valued functions in graph theory were studied, for example, in 4, 5, 6, 7, 8, 9, 10, 11, 13, 16, 17 . In Figure 1, a graph G and its point-block graph ^{Pb G} ( ) are shown.

G: Pb(G):

Figure 1

The block-point tree ^bP ( ) ^G of a connected graph G as the graph whose points can be put in one-to-one correspondence with the set of points and blocks of G in such a way that two points of ^bP ( ) ^G are adjacent if and only if 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 ³ . The block graph ^B ( ) ^G of a graph G _{is the} graph whose point set is the set of blocks of

G and two points are adjacent if the corresponding blocks contain a cut point of G in common. Clearly, ^Pb ( ) ^G contains both the block-point tree ^bP ( ) ^G and the block graph ^B ( ) ^G as disjoint subgraphs.

2. POINT-BLOCK GRAPHS

We start with a few preliminary remarks.

Remark 1: If v is a noncutpoint in G _, then it is an end point in ^Pb ( ) ^G .

Remark 2: If G is a connected graph, then

( ) ^G

Pb is also connected and conversely.

Remark 3: If G is a graph with p points and b blocks, then ^Pb ( ) ^{G has p} blocks and

b cutpoints and conversely.

The following will be useful in the proof of our results.

Theorem A ³ : If G is a connected graph with p points and b _i is the number of

blocks to which point v _i belongs in G _{, then} the block-point tree ^bP ( ) ^{G has} Σ b _i + 1 points and Σ b _i ^lines.

Theorem B ² : A graph H is the block graph of some graph if and only if every block of

H is complete.

When defining any class of graphs, if is desirable to know the number of points and lines in each; the first theorem determines the same.

Theorem 1: If G is a connected graph with p points and q lines and b _{i is the} number of blocks to which point v _{i belongs} in G , then the point-block graph ^Pb ( ) ^G

has Σ b _i + 1 points and Σ b _i ( b _i + ¹ ) ^{/ 2} _lines.

Proof: By Theorem A, if G is a connected graph with p points and b _i is the number of blocks to which point v _i belongs in G _{, then}

( ) ^G

bP _{has Σ + 1}

b i points and since the graphs ^bP ( ) ^G and ^Pb ( ) ^G have the same number of points, ^Pb ( ) ^G has Σ b _i + 1 points.

The number of lines of ^Pb ( ) ^{G is}

the sum of the number of lines in ^bP ( ) ^G

and in ^B ( ) ^G . By Theorem A, ^bP ( ) ^{G has}

b i

Σ lines. Also one easily verifies that

( ) ^G

B _has Σ ^{b b} _i ( _i − ^{1 / 2} ) lines. Hence the number of lines in ^Pb ( ) ^G

( − ¹ ) ^{/ 2}

Σ + Σ

= b _i b _i b _i

_{= Σ} ^b _i ( ^b _{i +} ¹ ) ^{/ 2} .

Corollary 1.1: If G is a graph with p points, q lines and m components and if b _i is the number of blocks to which point v _i belongs in G , then the point-block graph

( ) ^G

Pb _has b m

i +

Σ points and

( + ¹ ) ^{/ 2}

Σ b _i b _{i lines.}

Theorem 2: A graph G is a block if and only if ^Pb ( ) ^{G is} K 1 , , n ≥ ₂

n .

Proof: The direct part of the theorem is trivial.

On the other hand, suppose ^Pb ( ) ^G

is K 1 , , n ≥ 2

n , Then ^Pb ( ) ^G has a unique cutpoint and by Remark 3, G has a unique block. It implies that G is itself a block, which completes the proof of the theorem.

Theorem 3: A graph G is a totally disconnected graph if and only if the graphs

G _and ^Pb ( ) ^G are isomorphic.

Proof: The necessity is trivial.

For the sufficiency, suppose G _has a block. Then the number of points of G _is less than that in ^Pb ( ) ^{G . Hence} G ≠ ^Pb ( ) ^{G ,}

a contradiction. Thus G has no blocks and hence G is totally disconnected.

Theorem 4: A graph G is the point-block graph of some graph H if and only if

(i) Every block of G _is complete and (ii) Every block B _i ^has a unique noncutpoint v _i of degree n − _i 1 ^, where n _i is the number of points in B _i ^.

Proof: Suppose G = ^Pb ( ) ^{H ,} H is some graph. Then clearly every block of G _is complete and every block B _i ^of G _has unique noncutpoint v _i which corresponds to a point of H and also the degree of v _{i is}

1

i − n _.

Conversely, let (i) and (ii) be true for a graph G _{. Let} S denote the set of all cutpoints of G _{. Let}

H

be a graph obtained from G by removing the set of all points S . Clearly H _i is a totally disconnected graph with remaining points of G . Join those two points of H _i which are adjacent to a common cutpoint of G _, resulting a graph H . Clearly G = ^Pb ( ) ^{H .}

This completes the proof.

3. MIDDLE GRAPHS WITH POINT- BLOCK GRAPHS

The middle graph ^M ( ) ^G of a graph G is the graph whose point set is

( ) ^{G E} ( ) ^G

V ∪ and two points are adjacent if they are adjacent lines of G or one is a point and other is a line incident with it.

We now characterize those graphs whose point-block graphs and middle graphs are isomorphic.

Theorem 5: A connected graph G is a tree if and only if the graphs ^Pb ( ) ^{G and M} ( ) ^G

are isomorphic.

Proof: Suppose G is a tree. Then

( ) ^{G M} ( ) ^G

Pb = , since the lines and blocks

of a tree coincide.

Conversely, suppose ^Pb ( ) ^{G = M} ( ) ^G , G _is connected. If G has a cycle, then the number of blocks of G is less than the number of lines of G . It is known that the number of points of ^Pb ( ) ^G is the sum of the number of points and blocks of G _. Thus while ^M ( ) ^{G has p} + ^{q has less} number of points. Hence ^Pb ( ) ^{G ≠ M} ( ) ^{G ,}

a contradiction. Thus G _{has no cycles} and hence G is a tree.

4. RELATION BETWEEN POINT- BLOCK GRAPH AND LINE GRAPH

A graph G ⁺ is the endline graph of a graph G _if G ⁺ is obtained from G _by adjoining an endline u _i u _i ' at each point u _i of G . Hamada and Yoshimura have proved in ¹ that for any graph G _, M ( ) G = L ( ) G ⁺ _. The above consideration and Theorem 5 lead to:

Corollary 5.1: A connected graph G _{is a} tree if and only if Pb ( ) G = L ( ) G ⁺ _.

Theorem 6: If G is a block with p points, then L ( Pb ( ) G ) = K p ^.

Proof: Suppose a ( ^{p q ,} ) ^{graph G is a}

block. By Theorem 2, ^Pb ( ) ^{G is} K 1 . p . It is known that L ( K ^{1 .} _p ) = K _p ^{. Hence}

( ) ( Pb G ) K p

L = ^.

5. RELATION BETWEEN POINT- BLOCK GRAPH AND BLOCK GRAPH Theorem 7: If G is a nontrivial connected graph, then ^B ( ^Pb ( ) ^G ) = ^G if and only if

each block of G is complete.

Proof: Let G be a nontrivial connected graph. Suppose ^B ( ^Pb ( ) ^G ) = ^G . Then by Theorem B , each block of G is complete.

Conversely, suppose G is a graph with p points and b blocks and each block of G is complete. Then by Remark 3,

( ) ^G

Pb _{has p} blocks and b cutpoints.

Form, ^B ( ^Pb ( ) ^G ) ^{. Let G} ₁ be the resulting graph. Clearly, G _{1 has p} points and b blocks and by Theorem B , each block of

G 1 is complete. It implies that G _{1 and} G are isomorphic. Hence ^B ( ^Pb ( ) ^G ) = ^{G .} Theorem 8: For any graph G _,

( ) ^{G Pb ( ) G}

B ⁺ = ^.

Proof: Let ^b ( ) ^G be the set of blocks of G and ^b ( ) ^{G +} the set of blocks of G ⁺ _.

Let the elements { ^v _{i ,} ^u _i } , b _j of the set ^b ( ) ^{G +} correspond to the elements

{ } v i ^, b j _{of the set} ^V ( ) ^G _∪ ^b ( ) ^G , respectively. Then we have a one to one correspondence between the elements of the two sets. Hence we have a one to one correspondence between the points of the graphs ^B ( ) ^{G + and Pb} ( ) ^{G .}

By the assumption that the points u _{i are} different from the points v _{j of} G , and that

u i are distinct from each other, we have

{ v i u i } ∩ { v j u j } = { } v i ∩ { } v j = ^φ

, ,

( i ≠ j ) ,

{ ^v _i ^u _i } ∩ ^b _j = { } ^v _i ∩ ^b _j ^,

j i j

i b b b

b ∩ = ∩ ^.

This shows that the intersection of the to elements of ^b ( ) ^G coincides with that of the corresponding two elements of

( ) ^{G b} ( ) ^G

V ∪ . Hence we see that the adjacency of two points in ^B ( ) ^G coincides with that of the corresponding two points in

( ) ^G

Pb . Therefore, ^B ( ) ^{G +} = ^{Pb ( ) G . Thus} the theorem is proved.

6. RELATION BETWEEN POINT- BLOCK GRAPH AND BLOCK- POINT TREE

Theorem 9: If G is a block, then

( ) ^{G bP} ( ) ^G

Pb = ^.

Proof: Suppose a ( p, q ) graph G _{is a} block. Then the graphs ^Pb ( ) ^{G and} bP ( ) G have p + 1 points. By Theorem 2, ^Pb ( ) ^G

is K 1 . _p . Since G has only one block with p points, the point corresponding to this block is adjacent to p points in bP ( ) G . Thus ^{Pb G} ( ) is also _1, .

K p ^Hence

( ) ^{G bP} ( ) ^G

Pb = ^.

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