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The Point Block Graph of a Graph JOURNAL OF COMPUTER AND MATHEMATICAL SCIENCES ISSN 0976-5727 (Print) An International Open Free Access, Peer Reviewed Research Journal ISSN 2319-8133 (Online) Abbr:J.Comp.&Math.Sci. www.compmath-journal.org 2014, Vol.5(5): Pg.476-481 The Point Block Graph of A Graph V. R. Kulli and M. S. Biradar 1Department of Mathematics, Gulbarga University, Gulbarga, INDIA. 2Department 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 12 and was studied in 14,15 . Many other graph valued functions in graph theory were In this paper, we consider a graph a studied, for example, in 4, 5, 6, 7, 8, 9, 10, 11, 13, 16, 17 . finite, undirected without loops or multiple In Figure 1, a graph G and its 2 lines. We use the terminology of . point-block graph Pb( G ) are shown. Pb G The point-block graph ( ) 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 v G v other to a point of and is in B . This G: Pb (G): concept was introduce by Kulli and Biradar Figure 1 Journal of Computer and Mathematical Sciences Vol. 5, Issue 5, 31 October, 2014 Pages (412-481) 477 V. R. Kulli, et al., J. Comp. & Math. Sci. Vol.5 (5), 476-481 (2014) The block-point tree bP(G) of a blocks to which point vi belongs in G , then G connected graph as the graph whose the block-point tree bP(G) has Σbi +1 points can be put in one-to-one points and Σb lines. correspondence with the set of points and i blocks of G in such a way that two points Theorem B 2: A graph H is the block graph of bP G are adjacent if and only if one ( ) of some graph if and only if every block of corresponds to a block B of G and the H is complete. other to a point v of G and v is in B . When defining any class of graphs, This concept was introduced by Kulli 3. The if is desirable to know the number of points block graph B(G) of a graph G is the and lines in each; the first theorem graph whose point set is the set of blocks of determines the same. G and two points are adjacent if the Theorem 1: If G is a connected graph corresponding blocks contain a cut point of with p points and q lines and b is the G in common. Clearly, Pb(G) contains i number of blocks to which point v belongs both the block-point tree bP(G) and the i G Pb G block graph B(G) as disjoint subgraphs. in , then the point-block graph ( ) Σb +1 Σb b + 2/1 has i points and ( ii ) lines. 2. POINT-BLOCK GRAPHS G Proof: By Theorem A, if is a connected We start with a few preliminary graph with p points and bi is the number of remarks. blocks to which point vi belongs in G , then bP G Σb +1 Remark 1: If v is a noncutpoint in G , ( ) has i points and since the then it is an end point in Pb(G) . graphs bP(G) and Pb(G) have the same number of points, Pb(G) has Σbi +1 G Remark 2: If is a connected graph, then points. Pb(G) is also connected and conversely. The number of lines of Pb(G) is Remark 3: If G is a graph with p points the sum of the number of lines in bP(G) and b blocks, then Pb(G) has p blocks and and in B(G). By Theorem A, bP(G) has b cutpoints and conversely. Σbi lines. Also one easily verifies that Σb b − 1 / 2 The following will be useful in the B(G) has i( i ) lines. Hence the proof of our results. number of lines in Pb(G) 3 = Σb Σ+ b b − 2/1 Theorem A : If G is a connected graph i ( ii ) = Σb b + 2/1 with p points and bi is the number of ( ii ) . Journal of Computer and Mathematical Sciences Vol. 5, Issue 5, 31 October, 2014 Pages (412-481) V. R. Kulli, et al., J. Comp. & Math. Sci. Vol.5 (5), 476-481 (2014) 478 Corollary 1.1: If G is a graph with p Proof: Suppose G = Pb(H ), H is some points, q lines and m components and if bi graph. Then clearly every block of G is B G is the number of blocks to which point vi complete and every block i of has v belongs in G , then the point-block graph unique noncutpoint i which corresponds to Pb G Σb + m ( ) has i points and a point of H and also the degree of vi is Σb (bii + ) 2/1 lines. ni −1. Theorem 2: A graph G is a block if and Conversely, let (i) and (ii) be true Pb G K n ≥ 2 for a graph G . Let S denote the set of all only if ( ) is n,,1 . G H cutpoints of . Let i be a graph Proof: The direct part of the theorem is obtained from G by removing the set of all trivial. points S . Clearly H i is a totally On the other hand, suppose Pb(G) disconnected graph with remaining points of is K n n ≥ 2 , Then Pb(G) has a unique ,,1 G . Join those two points of H i which are G cutpoint and by Remark 3, has a unique adjacent to a common cutpoint of G , G block. It implies that is itself a block, resulting a graph H . Clearly G = Pb(H ). which completes the proof of the theorem. This completes the proof. G Theorem 3: A graph is a totally 3. MIDDLE GRAPHS WITH POINT- disconnected graph if and only if the graphs BLOCK GRAPHS G and Pb(G) are isomorphic. M G The middle graph ( ) of a graph Proof : The necessity is trivial. G is the graph whose point set is For the sufficiency, suppose G has V (G)∪ E(G) and two points are adjacent G a block. Then the number of points of is if they are adjacent lines of G or one is a less than that in Pb(G) . Hence G ≠ Pb(G), point and other is a line incident with it. a contradiction. Thus G has no blocks and We now characterize those graphs hence G is totally disconnected. whose point-block graphs and middle graphs are isomorphic. Theorem 4: A graph G is the point-block Theorem 5: A connected graph G is a tree H graph of some graph if and only if if and only if the graphs Pb(G) and M (G) G (i) Every block of is are isomorphic. complete and (ii) Every block Bi has Proof: Suppose G is a tree. Then a unique noncutpoint v of degree n −1, i i Pb(G) = M (G) , since the lines and blocks n B where i is the number of points in i . of a tree coincide. Journal of Computer and Mathematical Sciences Vol. 5, Issue 5, 31 October, 2014 Pages (412-481) 479 V. R. Kulli, et al., J. Comp. & Math. Sci. Vol.5 (5), 476-481 (2014) Conversely, suppose Pb(G) = M (G) , G is each block of G is complete. connected. If G has a cycle, then the G number of blocks of G is less than the Proof: Let be a nontrivial connected B Pb G = G number of lines of G . It is known that the graph. Suppose ( ( )) . Then by number of points of Pb(G) is the sum of Theorem B , each block of G is complete. G the number of points and blocks of G . Conversely, suppose is a graph with points and b blocks and each block Thus while M (G) has p + q has less p of G is complete. Then by Remark 3, number of points. Hence Pb(G) ≠ M (G), Pb(G) has p blocks and b cutpoints. a contradiction. Thus G has no cycles Form, B(Pb(G)). Let G1 be the resulting and hence G is a tree. graph. Clearly, G1 has p points and b 4. RELATION BETWEEN POINT- blocks and by Theorem B , each block of BLOCK GRAPH AND LINE GRAPH G1 is complete. It implies that G1 and G + A graph G is the endline graph of are isomorphic. Hence B(Pb(G)) = G . G G + G a graph if is obtained from by adjoining an endline uu ii ' at each point ui Theorem 8: For any graph G , + of G . Hamada and Yoshimura have proved B(G ) = Pb(G). + in 1 that for any graph G , M (G) = L(G ).
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