Block graph of a graph Текст научной статьи по специальности «Математика»

Владикавказский математический журнал 2019, Том 21, Выпуск 1, С. 74^78

ЗАМЕТКИ

УДК 519.17

DOI 10.23671/VNC.2019.1.27736

BLOCK GRAPH OF A GRAPH A. Kelkar1, К. Jaysurya1, Н. М. Nagesh1

1 Department of Computer Science and Engineering, P.E.S. Institute of Technology, Bangalore South Campus, Bangalore, Karnataka 560100, India E-mail: ashwinikelkar23@gmail. com, reddyjaysurya@gmail. com, [email protected]

Abstract. The block graph of a graph G, written B(G), is the graph whose vertices are the blocks of G

and in which two vertices are adjacent whenever the corresponding blocks have a cut-vertex in common.

We study the properties of B(G) and present the characterization of graphs whose B(G) are planar,

outerplanar, maximal outerplanar, minimally non-outerplanar, Eulerian, and Hamiltonian. A necessary

B(G)

Key words: crossing number, inner vertex number, dutch windmill graph, complete graph.

Mathematical Subject Classification (2010): 05C05, 05C45.

For citation: Kelkar, A., Jaysurya, K. and Nagesh, H. M. Block Graph of a Graph, Vladikavkaz Math.

J., 2019, vol. 21, no. 1, pp. 74-78. DOI: 10.23671/VNC.2019.1.27736.

1. Introduction

Notations and definitions not introduced here can be found in [1]. There are many graph operators (or graph valued functions) with which one can construct a new graph from a given graph, such as the line graph, the total graph, and their generalizations. One such generalization is the block graph concept whose properties and characterizations were considered in [2]. It is the object of this paper to study some of the structural properties of the block graph such as the planaritv, outer planaritv, etc.

We need some concepts and notations on graphs. A graph G = (V, E) is a pair, consisting of some set V, the so-called vertex set, and some subset E of the set of all 2-element subsets of V, the edge set. We write x = (p, q) and say that p and q are adjacent vertices (sometimes denoted p adj q).

A graph G is connected if between any two distinct vertices there is a path. A maximal connected subgraph of G is called a component of G. A cut-vertex of a graph is one whose removal increases the number of components. A non-separable graph is connected, nontrivial, and has no cut-vertices. A block of a graph is a maximal non-separable subgraph. If two distinct blocks B1 and B2 are incident with a common cut-vertex, then they are called adjacent blocks.

© 2019 Kelkar, A., Jaysurya, K. and Nagesh, H. M.

A graph G is planar if it has a drawing without crossings. For a planar graph G, the inner vertex number i(G) is the minimum number of vertices not belonging to the boundary of the

G

G

of the exterior region, then G is said to be outerplanar. An outerplanar graph G is maximal

G

minimally non-outerplanar if i(G) = 1 [3]. The least number of edge crossings of a graph G, among all planar embeddings of G, is called the crossing number of G and is denoted by cr(G).

A star graph K1>n (n ^ 3), is the complete bipartite graph. The dutch windmill graph Dgm\ also called a friendship graph, is the graph obtained by taking m copies of the cycle graph C3 with a vertex in common and therefore corresponds to the usual windmill graph D^. It is therefore natural to extend the definition to Dnm), consisting of m copies of Cn.

Definition 1.1. The line graph of a graph G, written L(G), is the graph whose vertices are the edges of G, with two vertices of L(G) adjacent whenever the corresponding edges of G have a common vertex.

Definition 1.2. The block graph of a graph G, written B(G), is the graph whose vertices G

have a cut-vertex in common.

Note that B(G) is defined only for graphs which have at least one cut-vertex or (at least two blocks). In Fig. 1, a graph G and its B(G) are shown.

B

3

Bi

Bi

Fig. 1.

B

B

4

2. Properties of Block Graphs

In this section we present some of the basic properties of B(G).

Property 2.1. If G is a tree of order n (n ^ 3), then L(G) = B(G).

Property 2.2. There is no non-trivial graph G which is isomorphic to its B(G).

Property 2.3. The block graph B(G) of a graph G is a block if G contains exactly one cut-vertex.

Property 2.4. If the number of cut-vertices of a path Pn (n ^ 3) is a, then number of cut-vertices of the corresponding B(Pn) is a — 1. Clearly, the number of cut-vertices of B(Ki,n) is zero.

Property 2.5. If G is a path Pn (n ^ 2), then the size of B(Pn) equals \ dj —n +1, where d is the degree of the vertices of Pn.

Property 2.6. If G is a star graph K\,n (n ^ 3), then the size of B(Ki>n) = ra(-ra2~ 1->.

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Kelkar, A., Jaysurva, K. and Nagesh, H. M.

3. Characterization of B(G) We now characterize the graphs whose B(G) are planar.

Theorem 3.1. The block graph B(G) of a graph G is planar if and only if G is either a star graph K1>n (2 ^ n ^ 4) or a dutch windmill graph D^ (2 ^ m ^ 4).

< Suppose that B(G) is planar. Assume that G = K1>n (n ^ 5). If G = K1>5, then B(G) = K5, which is non-planar, a contradiction. Assume now that G = Dinn) (m ^ 5). If G = D^, the n B (G) = K5, again a contradiction.

Conversely, suppose that G is either a star graph K1>n (2 ^ n ^ 4) or a dutch windmill graph Dlnm) (2 < m ^ 4). We consider the following cases. Case 1: If G = K1>2, then B(G) = K2, which is planar. Case 2: If G = K1>3, then B (G) = K3, which is planar. Case 3: If G = K1>4, then B(G) = K4, which is planar.

(2)

Case 4: If G = Dn , then B(G) = K2, which is planar. Case 5: If G = D^, then B(G) = K3, which is planar. Case 6: If G = D^, then B(G) = K4, which is planar.

Therefore, bv all the cases above, B(G) is planar. This completes the proof. >

B(G)

outerplanar; and minimally non-outerplanar.

B(G) G G

(3)

either K1>3 or Dn .

< Suppose that B(G) is outerplanar. Assume that G is either K1>n (n ^ 4) or Dnm) (m ^ 4). If G = K1)4, then B(G) = K4. Clearly the inner vertex number of B(G) is one, i.e., i(B(G)) = 1, a contradiction. Assume now that G = Dlm) (m ^ 4). If G = D{n4), then B(G) = K4, again a contradiction.

(3)

Conversely, suppose that G is either K1)3 or Dn ■ If G = K1>3, then B(G) = K3. Clearly

(3)

the inner vertex number of B(G) is zero, i. e., i(B(G)) = 0. If G = Dn , then B(G) = K3, and thus i(B(G)) = 0. Therefore, B(G) is outerplanar. This completes the proof. >

B(G) G

G K1 , 3 P 3

< Suppose that B(G) is maximal outerplanar. Assume that G is K1>n (n ^ 4). If G = K1>4, B(G) = K4 G Pn

of order n (n ^ 4). By definition, B(G) is a path of order n — 1. Clearly, i(B(G)) = 0, and

B(G) B(G)

not maximal outerplanar, again a contradiction.

Conversely, suppose that G is either K1>3 or a path P3. If G = K1>3, then B(G) = K3, which is maximal outerplanar. If G = P3, then B(G) = P2, which is also maximal outerplanar. This completes the proof. >

B(G) G

either K1>4 or D^.

< Suppose B(G) is minimally non-outerplanar. Assume that G = K1>5. Bv definition, B(G) = K5, which is non-planar, a contradiction. On the other hand, if G = D^, then B(G) = K5

Conversely, suppose that G is either K1>4 or D^. By definition, B(G) = K4. Clearly, i(B(G)) = 1. Hence B(G) is minimally non-outerplanar. This completes the proof. >

Theorem 3.5. The block graph B(G) of a graph G has crossing number one if and only if G is either K1>5 or D(5).

< Suppose G has crossing number one. Assume that G = K1>n (n ^ 6). If G = K1)6, then B(G) = K6. Clearlv, cr(B(G)) > 1, a contradiction. On the other hand, if G = D^, then B(G) = K6, a contradiction.

Conversely, suppose that G is either K1)5 or D^. By definition, B(G) = K5. Since the crossing number of K5 is exactly one, cr(B(G)) = 1. This completes the proof. >

Definition 3.1. An Eulerian cycle in an undirected graph is a cycle that uses each edge exactly once. If such a cycle exists, then the graph is called Eulerian.

Theorem 3.6 (Hararv [1]). A connected graph G is said to be Eulerian if and only if the G

Theorem 3.7. The block graph B(G) of a graph G is Eulerian if and only if G is either K12k+1 or D^^1"1 (k ^ 1).

< Suppose B (G) is Eulerian. Assum e that G = K1;2k (k ^ 1) By definiti on, B (G) = K2k in which degree of each vertex is 2k — 1, which is odd. Since the degree of each vertex of B(G) is odd, Theorem 3.6 implies that B(G) is non-Eulerian, a contradiction. On the other hand, if G = Dn2k), then B(G) = K2fc, again a contradiction.

Conversely, suppose that G is either K1;2k+1 or Dn2fc+1) (k ^ 1). By definition, B(G) = K2k+1, in which the degree of each vertex of B(G) is 2k which is even for every k ^ 1. Since the degree of each vertex of B(G) is even, Theorem 3.6 imp lies that B(G) is Eulerian. This completes the proof. >

Definition 3.2. A Hamiltonian path is a path that visits each vertex of the graph exactly one. A graph is Hamiltonian if for every pair of vertices there is a Hamiltonian path between the two vertices.

Theorem 3.8. The block graph B(G) of K1>n (n ^ 3) or Dt" (m ^ 3) is Hamiltonian.

< Suppose that G is either K1>n (n ^ 3) or D^ (m ^ 3). By definition, B(G) is a complete graph of order n or m. Since every complete graph is Hamiltonian, B(G) is Hamiltonian. This completes the proof. >

4. Open problems

4.1. One can naturally extend these concepts to the directed graph version. What can one say about the properties of the directed version?

4.2. If the number of cut-vertices of the graph G is then what is the number of cut-vertices of the corresponding B(G)?

References

1. Harary, F. Graph Theory, Reading, Addison Wesley, 1969.

2. Harary, F. A Characterization of Block-Graphs, Canadian Mathematical Bulletin, 1963, vol. 6, issue 1, pp. 1-6. DOI: 10.4153/CMB-1963-001-X.

3. Kulli, V. R. On Minimally Nonouterplanar Graphs, Proceeding of the Indian National Science Academy, 1975, vol. 40, pp. 276-280.

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Kelkar, A., Jaysurva, K. and Nagesh, H. M.

Received, May 7, 2018 Ashwini Kelkar

Department of Computer Science and Engineering, P.E.S. Institute of Technology, Bangalore South Campus, Bangalore, Karnataka 560100, India Student

E-mail: ashwinikelkar23@gmail. com

K. Jaysurya

Department of Computer Science and Engineering, P.E.S. Institute of Technology, Bangalore South Campus, Bangalore, Karnataka 560100, India Student

E-mail: [email protected]

Hadonahalli M. Nagesh

Department of Computer Science and Engineering, P.E.S. Institute of Technology, Bangalore South Campus, Bangalore, Karnataka 560100, India Assistant Professor E-mail: [email protected]

Владикавказский математический журнал 2019, Том 21, Выпуск 1, С. 74^78

ГРАФ БЛОКОВ Келкар Э.1, Джейсурья К.1, Нагеш X. М.1

1 Кафедра компьютерной науки и техники, Технологический институт PES, Бангалор, Индия E-mail: ashwinikelkar23@gmail. com, [email protected], [email protected]

Аннотация. Граф блоков B(G) графа G — граф, вершинами которого являются блоки графа G и в котором две вершины смежны тогда и только тогда, когда соответствующие им блоки имеют общую точку сочленения. Изучаются различные свойства графа блоков B(G), в частности, даны характеристики графов, у которых графы блоков B(G) являются плоскими (планарными), внешнепланарными, максимальными внешнепланарными, минимальными невнешнепланарными, эйлеровыми и гамильтоно-выми. Также представлено необходимое и достаточное условие, чтобы число пересечения графа блоков B(G) равнялось единице.

Ключевые слова: число пересечения, число внутренних вершин, граф «голландская мельница», полный граф.

Mathematical Subject Classification (2010): 05С05, 05С45.

Образец цитирования: Kelkar, A., Jaysurya, К. and Nagesh, Н. М. Block Graph of a Graph // Владишвк. мат. журн, 2019.-Т. 21, № l.-C. 74-78 (in English). DOI: 10.23671/VNC.2019.1.27736.