SEMITOTAL AND TOTAL BLOCK-CUTVERTEX GRAPH

Transcription

1 CHAPTER 3 SEMITOTAL AND TOTAL BLOCK-CUTVERTEX GRAPH ABSTRACT This chapter begins with the notion of block distances in graphs. Using block distance we defined the central tendencies of a block, like B-radius and B-diameter of a block in a graph. An algorithm is given to find the block center of the graph. It is proved that block center of any graph is B-complete. Later six new graphs are defined viz - semitotal block cutvertex graph, total block cutvertex graph, semitotal block vertex graph, total block vertex graph, semitotal block vertex edge graph and total block vertex edge graph arising from the given graph. Expressions for number of edges in the newly defined graphs are derived.

2 3.1 Introduction For standard terminologies we refer F. Harary [1]. Distance plays a prominent role in the study of Graph Theory. For any two vertices a, b in G the distance d ( a, b) is the length of a shortest path between a and b. Let x, y X, a V and x ( u, v) then the distance between the vertex a and an edge x, is defined as d( x, a) min{ d( u, a), d( v, a)} and the distance between the two edges d( x, y) min{ d( y, u), d( y, v)}. The eccentricity of a vertex v V(G) is defined as. Then diameter and radius. For any graph G,. The set of vertices with minimum eccentricity is called center C (G) of the graph. For any tree T the center consists of a single vertex or two adjacent vertices. Using distances, several problems are posed in graph theory such as Chinese postman problem and shortest distance problem etc. Many well known algorithms like Dijikstra s algorithm, Kruskal s algorithm and Prim s algorithm have been generated using the distance concepts. Similar to distance in graphs, block distance in graphs are defined. 3.2 B-distance between two blocks B-paths are defined in Chapter 2. We recall that the length of a B-path is the number of cutvertices in it. Definition For any two blocks b 1, b 2 B(G), the B-distance d(b 1, b 2 ) is the length of the B-path from b 1 to b 2. Further, for any b 1, b 2, b 3 B(G) the block distance has the following properties. 42

3 (i) d(b 1, b 2 ) 0 and d(b 1, b 2 ) = 0 if and only if b 1 = b 2. (ii) d(b 1, b 2 ) = d(b 2, b 1 ) (iii) d(b 1, b 2 ) d(b 1, b 3 ) + d(b 3, b 2 ) - Triangular inequality. Therefore B-distance is a metric. Using B-distance the central tendencies of a block, like B-radius and B-diameter of a block in a graph can also be defined. Definition The B-eccentricity of a block h B(G) is defined as. Then B-diameter and B-radius. The set of blocks with minimum eccentricity is called B-center C b (G) of the graph. Immediately these parameters yield a lower bound for the usual diameter d(g) and radius r(g) of a graph. Hence we have and. Proposition Let G be a graph with B-diameter d b (G) and B-radius r b (G). Let and be the block graph and cutvertex graph of G. Then (3.1) ( ) (3.2) ( ) (3.3) ( ) (3.4) ( ) (3.5) Proof. Equation (3.1) follows from the fact that. To prove (3.2). Consider the diametrical B-path in G. 43

4 Then = number of cutvertices in the diametrical B-path in G = number of blocks in the diametrical B-path in B G (G) = number of cutvertices in the diametrical B-path in B G (G) + 1 = ( ). Since B G (B G (G)) = C G (G) equation (3.3) follows from (3.2). To prove (3.4). Again from (3.1) and (3.2), we have ( ( ) ) ( ) ( ) as desired. Also from (3.3) we have, ( ). Hence ( ) ( ) which yields (3.5). Recall that a skeleton of G is obtained by deleting all the pendant blocks from G. An i th skeleton S i (G) is obtained iteratively by getting skeleton of G, i times. Therefore G = S 0 (G). Definition If a vertex v is a cutvertex both in G and S 1 (G) then v is a non-end-cutvertex of G, otherwise v is an end-cutvertex of G. Proposition (i) (ii) If G has atleast one cutvertex then there exists atleast two pendant blocks. If G has atleast two cutvertices then there exists atleast two end-cutvertices. Proof. Consider the diametrical B-path in G. Then the two blocks with maximum B-eccentricity are the required two pendant blocks and the cutvertex incident on these two blocks are the required two end-cutvertices. Proposition For any graph G with m blocks the B-center C b (G) is B-complete. Proof. Case 1. Suppose m = 1. Then d b (G) = r b (G) = 0. Hence C b (G) = B 1 and the result follows. 44

5 Case 2. Suppose m >1 and G = B m Then d b (G) = r b (G) = 1. Hence C b (G) = B m and the result follows. Case 3. Suppose m >1 and G is not a B-complete graph. Then G has atleast two cutvertices. Then by Proposition there exists atleast two pendant blocks. Now obtain the skeleton S 1 (G). Then all the blocks are at B-distance one less than the B- distance in the original graph. Also the blocks which were in the B-center in the original graph remain same in the new graph. Hence C b (G) = C b (S 1 (G)). Let k 1 be the number of blocks in S 1 (G). Check whether S 1 (G) is B-complete and if so, from case 1 and case 2, the result follows and we stop. If S 1 (G) is not B-complete, we get S 2 (G). The process of getting skeleton S i +1 (G) from S i (G) is repeated, if and only if S i (G) is not B-complete. After say n steps we get S n (G) which is B-complete with k n blocks and C b (G) = C b (S n (G)) = S n (G) which is B-complete. Then by case 1 and case 2, the result follows. This completes the proof. From the above proof it is possible to get an algorithm to find B-centre of G which eventually gives the B-radius of G. Note that if we take skeleton of G, i times implies that block radius of G is i or i + 1 according as B-center is B 1 or B m where m >1. The algorithm proceeds as follows. Algorithm i = 0 count = m k = count begin while (S i (G) B k ) S i (G) = S i+1 (G) i = i + 1 count = number of blocks in S i+1 (G) 45

6 end k = count end while C b (G) = S i (G) if k = 1 block radius of G = i else block radius of G = i+1 end if 3.3 Semitotal-block-cutvertex graph and total-block-cutvertex graph. The line graph L(G) of a graph G is a graph with vertex set X(G) and any two vertices in L(G) are adjacent if and only if the corresponding lines are adjacent in G. The block graphs are studied in [70 and 72]. The edges and vertices of G are called its members. Semitotal graph t(g) is a graph as defined by Sampathkumar and Chikodimath [73] with vertex set X(G) V(G) and any two vertices in t (G) are adjacent if and only if the corresponding vertices are adjacent in G or the corresponding members are incident. The total graph T(G) introduced by M. Behzad [74] is a graph with vertex set X(G) V(G) and two vertices in T(G) are adjacent if and only if the corresponding members are adjacent or incident in G. V.R.Kulli [75] defined and studied the properties of semitotal- block graph and total-block graphs. The vertices and blocks of a graph are called its members. Semitotal-block graph T b (G) of a graph G is a graph with vertex set B(G) V(G) and two vertices are adjacent if and only if the corresponding vertices are adjacent in G or corresponding members are incident. The total block graph T B (G) has vertex set B(G) V(G) and two elements in T B (G) are adjacent if and only if they are adjacent or incident to each other in G. Similar to block cutvertex graph, V.R.Kulli [75] defined the block 46

7 vertex tree of a graph G, denoted as bp(g) which is a tree with vertex set B(G) V(G) and a vertex v V (G) and a block b B (G) are adjacent if and only if v is incident with the block b. Motivated by these definitions we now define two new graphs arising from the given graph. Definition The blocks and cutvertices of a graph are called its members. Semitotal block cutvertex graph T bc (G) of a graph G is a graph with vertex set B(G) C(G) and any two vertices in T bc (G) are adjacent if and only if the corresponding cutvertices are adjacent or the corresponding members are incident. It is immediate that T bc (G) = BC(G) C G (G). Definition The total block cutvertex graph T BC (G) of a graph G is a graph with vertex set B(G) C(G) and any two vertices in T BC (G) are adjacent if and only if the corresponding members are adjacent or incident. Again we note that T BC (G) = BC(G) C G (G) B G (G). It is important to recall the following definitions defined in Chapter 2 to prove the next proposition. A block b is a pendant block if it is incident with only one cutvertex, otherwise b is a nonpedant block. Let B P (G) and B NP (G) denote the set of all pendant and nonpendant blocks of G respectively. Let m P = B P (G) and m NP = B NP (G). B NP (G) B P (G) =B(G) and B NP (G) B P (G) =, hence m = m p + m NP. Proposition For any graph G, with m NP pendant blocks and n cutvertices, (3.6) ( ) ( ) (3.7) 47

8 Proof. From Proposition 2.4.2, we have. Since each pendant block (has only one cutvertex) contribute one to the sum of cutvertex degrees, we have. Then Now, ( ) ( ) ( ) ( ). Hence the proof. An expression for the number of edges in the semitotal and total block cutvertex graph is derived in the next Theorem. Henceforth, by we mean the number of edges in. Theorem Let G be a graph with m blocks and n cutvertices. Let p bc and q bc denote the number of vertices and number of edges in T bc (G). Let p BC and q BC denote the number of vertices and number of edges in T BC (G). Then (i) p bc = m + n = p BC (3.8) (ii) ( ( ) ) (3.9) (iii) [ ( ) ( ) ] (3.10) Proof. Since the vertex set of T bc (G) and T BC (G) being the disjoint union of B(G) and C(G) we have p bc = p BC = B(G) + C(G) = m + n. To prove equation (3.9). Since every cutvertex yields d vb (c) edges in BC(G), we have q(bc(g)) =. Further, since all the cutvertices incident to a 48

9 nonpendant block are mutually adjacent, every nonpendant block h yields ( ) edges in C G (G). Hence ( ) ( ). Then, ( ) ( ) ( since T bc (G) = BC(G) C G (G) ( ) ( ) ( ) ( ) (using Proposition 3.3.3) ( ) ( ( ) ) (since m- m NP = m p ). ( ( ) ) ( ( ) ). To prove equation (3.10). Since all the blocks incident to a cutvertex c are mutually adjacent, every cutvertex yields ( ) edges in B G (G). Hence ( ) ( ). Then, ( ) ( ) ( ) (since ) ( = ) ( ) [ ( ) ( ) ] 49

10 (using Proposition 3.3.3) [ ( ) ( ) ] (Since from Theorem 2.4.2) [ ( ) ( ) ] (since )). Example For the graph G in the Fig.3.3.6, ( ) ( ( ) ) which in agreement with the result obtained in the Theorem Similarly, [ ( ) ( ) ] which satisfies the result of the Theorem T T bc (G) BC (G) G Fig Semitotal and Total block- cutvertex graph of G 3.4 Semitotal-block-vertex graph and total-block-vertex graph Definition The blocks and vertices of a graph are called its members. Semitotal-blockvertex graph T bp (G) is a graph with vertex set V(G) B(G) and any two vertices in T bp (G) are adjacent if and only if the corresponding vertices are vv-adjacent or the corresponding members are incident. It is immediate that T bp (G) = b p (G) P G (G). Definition The total-block-vertex graph T BP (G) is a graph with vertex set V(G) B(G) and any two vertices in T BP (G) are adjacent if and only if the corresponding members are vv-adjacent or adjacent or incident. Again we note that T BP (G) = b p (G) P G (G) B G (G). 50

11 It is obtained an expression for the number of edges in the semitotal-block-vertex and total-block-vertex graphs in the next result. Theorem Let G be a (p, q) graph with m blocks and n cut-vertices. Let q bp and p BP denote the number of edges in T bp (G) and T BP (G) respectively. Then Proof. To prove equation (3.11). ( ( ) ) (3.11) [ ( ) ( ) ] (3.12) Since every vertex yields d bv (h) edges in b p (G), ( ). (from Theorem 2.4.3). Then, ( ) ( ) [ ( ) ] [ ( ) ]. To prove equation (3.12). ( ) ( ) ( ) [ ( ) ] [ ( ) ] [ ( ) ] [ ( ) ] [ ( ) ( ) ] (using Theorem and Theorem 2.4.3) 51

12 Example For the graph G in the Fig.3.4.5, ( ( ) ) which in agreement with the result obtained in the Theorem Similarly, [ ( ) ( ) ] which satisfies the result obtained in Theorem G T bp (G) Fig Semitotal and total block- vertex graph of G T BP (G) 3.5 Semitotal-block-vertex-edge graph and total-block-vertex-edge graph Definition The blocks, vertices and edges of a graph are called its members. The Semitotal-block-vertex-edge graph T bpe (G) is a graph with vertex set V(G) B(G) X(G) and any two vertices in T bpe (G) are adjacent if the corresponding vertices are vv-adjacent or corresponding members are incident. Definition A vertex edge graph V e (G) is a bigraph with vertex set as V(G) X(G) and a vertex v V (G) and an edge x X(G) are adjacent in V e (G) if and only if v is incident on the edge x. 52

13 Definition A block edge graph b e (G) is a bigraph with vertex set as B(G) X(G) and a block b B (G) and an edge x X(G) are adjacent in b e (G) if and only if the edge x is incident on the block b. It is immediate that T bpe (G)= P G (G) b e (G) V e (G) b p (G). Definition The total-block-vertex-edge graph T BPE (G) is a graph with vertex set V(G) B(G) X(G) and any two vertices in T BPE (G) are adjacent if the corresponding members are vvadjacent or adjacent or incident. It is immediate that T BPE (G)= T bpe (G) B G (G) L(G). We note that the number of vertices in T BPE (G) and T bpe (G) is equal to p + q + m. graphs. In the next theorem we obtain an expression for number of edges in the above defined Theorem Let G be a (p, q) graph with m blocks and n cut-vertices. Let q bpe and p BPE denote the number of edges in T bpe (G) and T BPE (G) respectively. Then ( ( ) ) (3.13) [ ( ) ( ) ( ) ] (3.14) Proof. To prove (3.13). Since each edge is incident on two vertices, there are 2q edges in vertex edge graph V e (G). As every edge is incident on a unique block, there are q edges in block edge graph b e (G). Then, ( ) ( ) ( ) ( ) [ ( ) ] [ ( ) ] 53

14 To prove equation (3.14) We know that ( ) [ ( ) ] (see Harary [1]). Then, Example ( ) ( ) [ ( ) ] [ ( ) ( )] [ ( ) ] [ ( ) ( ) ( ) ] V 2 G V 1 T bpe (G) T BPE (G) Fig Semitotal and total block- vertex- edge graph of G For the graph in the Fig.3.5.7, ( ) ( ) ( ) ( ). Therefore Again ( ) ( ( ) ) which satisfies the result obtained in the Theorem Similarly, ( ) ( ) Also [ ( ) ( ) ]. Hence the Theorem is verified. 54