The Lower and Upper Forcing Geodetic Numbers of Block-Cactus Graphs

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(1)The Lower and Upper Forcing Geodetic Numbers of Block-Cactus Graphs ∗ Fu–Hsing Wang1,2 1. Yue–Li Wang1,∗†. Department of Information Management, National Taiwan University of Science and Technology, Taipei, Taiwan, Republic of China. 2. Department of Information Management, Chungyu Institute of Technology, Keelung, Taiwan, Republic of China.. Abstract. 1. Introduction All graphs considered in this paper are fi-. A vertex set D in graph G is called a geodetic set if all vertices of G are lying on some shortest u − v path of G, where u, v ∈ D. The geodetic number of a graph G is the minimum cardinality among all geodetic sets. A subset S of a geodetic set D is called a forcing subset of D if D is. nite and simple (i.e., without loops and multiple edges). Let G = (V, E) be a graph with vertex set V and edge set E. The cardinalities of V and E are called order and size, respectively. The distance of two vertices u and v in a connected graph G, denoted by d(u, v), is the number of edges in a. the unique geodetic set containing S. The forcing. shortest u − v path. A shortest u − v path is also. geodetic number of D is the minimum cardinal-. called a u − v geodesic. Let I(u, v) denote the set. ity of a forcing subset of D, and the lower and. of all vertices lying on some u − v geodesic of G, S and I(S) = u,v∈S I(u, v), where S ⊆ V (G).. upper forcing geodetic numbers of a graph G are the minimum and maximum forcing geodetic numbers, respectively, among all geodetic sets of G. In this paper, we find out the lower and upper forcing geodetic numbers of block-cactus graphs. Keyword: Geodetic set, Block-cactus graphs.. A vertex set D in graph G is called a geodetic set if I(D) = V (G). A geodetic set with the minimum cardinality is said to be a minimum geodetic set (gset for short). The cardinality of a g-set, denoted g(G), is called the geodetic number of a graph G [12]. Notice that 2 ≤ g(G) ≤ |V | for |V | ≥ 2. In [2], Buckley et al. characterized those connected. ∗ This. research was partially supported by National Science Council under the Grant NSC91-2213-E-011-044. † All correspondence should be addressed to Professor Yue–Li Wang, Department of Information Management, National Taiwan University of Science and Technology, 43, Section 4, Kee–Lung Road, Taipei, Taiwan, Republic of China. (Email: ylwang@cs.ntust.edu.tw).. graphs for which the geodetic number is equal to |V |, |V | − 1, or 2. Two classes of graphical games called achievement and avoidance games were examined for the geodetic number[1, 10]. Lately,.

(2) Chartrand et al. boost the research on geodetic set. a. problems [4, 5, 6, 7, 8] and determine the geodetic. b. c. d. e. numbers for cycles, trees, etc. [8]. However, determining the geodetic number of a general graph is NP-hard [9].. Figure 1: A graph G with g(G) = 3.. A subset S of a g-set D is called a forcing subset of D if D is the unique g-set containing S. This means that D can be figured out after S is determined. A vertex in S is said to be a forcing vertex of D. The forcing geodetic number of D, denoted f (D), is the minimum cardinality of a forcing subset for D [6]. The upper forcing geodetic number, denoted f + (G), of a graph G is the maximum forcing geodetic number among all gsets of G [12]. Notice that f + (G) = 0 if and only if G has exactly one g-set. In contrast, we define the lower forcing geodetic number, denoted f − (G),. Researches on forcing concepts have been widely studied such as forcing domination number [3], forcing perfect matching [11] and forcing geodetic number [6].. termined the upper forcing geodetic numbers for trees, cycles, complete bipartite graphs and hypercubes [12]. In this paper, we furthermore find out the lower and upper forcing geodetic numbers of block-cactus graphs which are the general case of cycles and trees.. of a graph G to be the minimum forcing geodetic number among all g-sets of G. We use Figure 1 as an example to illustrate the above notation. In Figure 1, vertex set {a, b, c, d, e} is intuitively a geodetic set of G. There are only three g-sets in G, namely D1 = {a, b, e}, D2 = {a, c, d} and D3 = {a, d, e}. Thus, g(G) = 3. Since D1 is the only g-set containing b, it follows that f (D1 ) = 1.. Recently, Zhang de-. The remaining part of this paper is organized as follows. The next section introduces some basic terminologies, notation and previous results. In Section 3, we study the problem of finding the lower and upper forcing geodetic numbers on block-cactus graphs. Finally, we give concluding remarks and address our future researches in the last section.. Furthermore, D2 is the only g-set containing c. Thus, f (D2 ) = 1. In D3 , since every vertex of D3 is also contained in some other g-set, f (D3 ) ≥ 2.. 2. Preliminaries and Previous Results. It can be found that D3 is the unique g-set containing {d, e}, and hence f (D3 ) = 2.. There-. For any set S of vertices in a graph G, the sub-. fore, f (G) = min{f (D1 ), f (D2 ), f (D3 )} = 1 and. graph induced by S, denoted by [S], is the maxi-. f + (G) = max{f (D1 ), f (D2 ), f (D3 )} = 2. More-. mal subgraph of G with vertex set S. The induced. over, the forcing number of a disconnected graph. subgraph [V \S] is denoted by G−S. It is the sub-. is defined to be the sum of the forcing numbers. graph obtained from G by deleting the vertices in. of all components. For simplicity, all the graphs. S together with their incident edges. If S = {v},. considered in this paper are connected.. then we write G−v for G−{v}. For a g-set D, the. −.

(3) 9. 22. B5. 2. 8. 6. 7. 1 3 5 16. 7. 4 10. 15. 5. 9. 14. B3 15. 13. B7. 11. 14 11. B2. 3. 4. 10. 2. B1. 6. 8. B6. 1. 20. 12 16. 13. B4. 19. B8. 12. 21. 17. 18. Figure 2: A block graph. Figure 3: A cactus graph G. contribution of [S] to f (D) is the cardinality of a 22. B5. 8. subset of S which is also a forcing subset of D. A vertex v is called an extreme vertex if the subgraph induced by the neighbors of v is complete. A ver-. 7. 6 5. 4. 14 11. B2. B3 15. 13 20. B8. 10. 3 B7. edges incident to it increases the number of com-. without a cut vertex. A graph G is called a block. 9. 2. B1. tex v is called a cut vertex if removing v and all. ponents. A block of a graph is a maximal subgraph. B6. 1. 16 B4. 19. 21. 12. 17. 18. Figure 4: A block-cactus graph.. graph if and only if every block of G is complete. Clearly, every vertex of a block graph is either a cut vertex or an extreme vertex. Figure 2 depicts. cut vertices are 2, 3, 7, 11, 19 and 20. Blocks B1 , B2. a block graph in which vertices 3, 4, 5, 8 and 10. and B4 are CIBs, while B3 is a CEB. The orders. are cut vertices while other vertices are extreme. of B1 , B2 , B3 and B4 are 7, 6, 3 and 5, respec-. vertices.. tively. Blocks B1 , B3 and B4 are odd and B2 is. A block that is a cycle is called a cyclic block. A. even. Figure 4 is a block-cactus graph. We can. cyclic block B is odd (respectively, even) if the or-. see that B2 and B4 in Figure 3 are changed to. der of B is odd (respectively, even). A cyclic block. complete graphs in Figure 4.. with a unique cut vertex is called a cyclic end-block. Consider CIBs of G and let B be a CIB of G.. (CEB for short). We also call a block with more. A path P in B is called a segment if both of its. than one cut vertex a cyclic internal-block (CIB. end vertices are cut vertices and the other vertices. for short). A cactus graph is a graph in which ev-. in P are not cut vertices. Clearly, |P | < n. A seg-. ery block with three or more vertices is a cyclic. ment P is said to be a long segment if the length. block.. of P is greater than. Furthermore, a graph whose blocks are. n 2.. Intuitively, B has at most. either cycles or complete is called a block-cactus. one long segment while other segments are called. graph. Block-cactus graphs generalize the known. short segments. A CIB having a long segment is. classes of block graphs and cactus graphs [13]. For. also called an LCIB except that the size of the CIB. example, Figure 3 illustrates a cactus graph G. is odd and the length of the long segment is one. with blocks {B1 , B2 , B3 , B4 , B5 , B6 , B7 , B8 } and. less than the size of the CIB. We use l(G) to de-.

(4) note the number of LCIBs in G. Take Figure 3 as. For a complete graph or a tree G, G has ex-. an example. Block B1 has three segments in which. actly one g-set which consists of all extreme ver-. path: 3, 4, 5, 6, 7 is a long segment while other seg-. tices. Thus, by Corollary 5, we have the following. ments are short. So, B1 is an LCIB. Moreover, B4. corollary.. is another LCIB while B2 is not. In B4 , segment S1 : 19, 20 and S2 : 20, 16, 17, 18, 19 are with the. Corollary 6 If G is a complete graph or a tree, then f − (G) = 0.. same end vertices. Nevertheless, S1 is short and S2 is long. Furthermore, since the length of S2 is. 3. The Forcing Geodetic Numbers of Block-cactus Graphs. equal to the order of B4 minus 1 and S2 is in an odd CIB, S2 is not an LCIB. Therefore, l(G) = 1. The following lemmas shown by Zhang [12] are. For the lower forcing geodetic number f − (G) of. helpful to clarify our proof for determining the. a graph G, if f − (G) = 0, then G has exactly one. lower forcing geodetic numbers of other graphs.. g-set. Thus, the next lemma follows directly from Lemma 1.. Lemma 1 For a graph G, f + (G) = 0 if and only Lemma 7 The following statements are equiva-. if G has exactly one g-set.. lent: For an integer k ≥ 2, Zhang [12] showed that g(C2k ) = 2 and g(C2k+1 ) = 3, where Cn is a cycle of n vertices. The next lemma describes the upper forcing geodetic numbers for  0    1 Lemma 2 f + (Cn ) = 2    3. (1) f + (G) = 0, (2) G has exactly one g-set, (3) f − (G) = 0.. Cn . if if if if. n = 3, n is even, n = 5, n ≥ 7 is odd.. The next two lemmas state the inclusion and exclusion of extreme vertices and cut vertices, respectively, with respect to geodetic sets. Lemma 3 Every extreme vertex belongs to every geodetic set. Lemma 4 If w is a cut vertex, then w cannot be. For a block graph G, the geodetic number of G can be obtained directly from Lemmas 3 and 4. Furthermore, the lower and upper forcing geodetic numbers of G follow from Lemma 4 and Corollary 5. Theorem 8 Let c be the number of cut vertices of a block graph G. Then, g(G) = |V | − c and f − (G) = f + (G) = 0. To determine the forcing geodetic numbers on cactus graphs, we need to find out the forcing. a vertex of any g-set.. geodetic numbers of cycles. By Lemma 3, every extreme vertex is certainly contained in every g-set.. Therefore, the next. corollary follows. Corollary 5 Every extreme vertex cannot be a forcing vertex.. In contrast with. Lemma 2, we show the lower forcing geodetic number of a cycle as follows. Lemma 9 f − (Cn )   0 if n = 3, 1 if n is even,  2 if n is odd and n 6= 3.. =.

(5) If n = 3, then every vertex is obviously. one vertex, say u, in D. Let Dv = D ∩ V (B) ∪ {v}.. an extreme vertex. By Lemma 3, f − (C3 ) = 0. For. Then, Dv is clearly a g-set of B. If the order. an even cycle Cn : v0 , v1 , . . . , v2k−1 , v0 , every g-set. of B is even, then, by Lemmas 2 and 9, B has. is of the form {vi , v(i+k) mod 2k } which is the only. exactly one forcing vertex in Dv . We can adjust. g-set containing vi , where 0 ≤ i ≤ 2k − 1. Thus,. this forcing vertex to be vertex v and this lemma. f − (Cn ) = 1 for n is even.. follows. Now, we consider that the order of B is. Proof.. At first, we show that f − (Cn ) ≥ 2 for the case. odd. Let S denote a forcing subset of Dv . Since. where n is odd and n 6= 3. It is clear that there. the order of B is greater than 3, by Lemmas 2. are more than one g-set in Cn . By Lemma 7,. and 9, |S| ≥ 2. Without loss of generality, we. f − (Cn ) > 0. We now show that f − (Cn ) 6= 1.. can adjust S so that v is a vertex in S. Note that. Suppose to the contrary that there exists a g-set. the vertices in S \ {v} are still forcing vertices of. such that f − (Cn ) = 1.. D. This implies that B contributes f − (B) − 1 to. With vertex symmet-. ric property of cycles, let D = {v0 , vi , vj }, 0 ≤. f − (G) and f + (B) − 1 to f + (G).. i, j ≤ 2k − 1, be the unique g-set containing v0 .. Q. E. D.. This implies that v0 cannot be contained in any. The following notation will be used in Lem-. other g-set. Nevertheless, both {v0 , v1 , vk+1 } and. mas 11 and 12. Let P be a segment in CIB B. {v0 , vk , vk+1 } are also g-sets. It is a contradiction.. of cactus graph G and the end vertices of P be. 0. 0. Finally, we present a g-set D with f (D ) = 2. a and b. Since a (respectively, b) is a cut ver-. to complete the proof. Let D0 = {v0 , v1 , vk+1 }. tex, G − a (respectively, G − b) has at least two. be a g-set of Cn : v0 , v1 , . . . , v2k , v0 . Clearly, D. components. Let Ga (respectively, Gb ) denote the. is the unique g-set containing {v0 , v1 } and hence. subgraph which consists of all the components of. −. f (D) = 2. We conclude that f (Cn ) = 2 if n is. G − a (respectively, G − b) except the component. odd and n 6= 3.. containing P . It can be seen that G consists of Q. E. D.. Ga , Gb and B.. Consider a CEB B of cactus graph G and let w be the cut vertex of B. If the order of B is 3,. Lemma 11 If w is a vertex of a short segment,. then all the vertices in V (B) \ {w} are clearly ex-. then w cannot be a vertex of any g-set.. treme vertices, and hence are not forcing vertices. Proof.. Let P be a short segment in cycle B of. Therefore, We only need to consider the CEBs of order ≥ 4.. cactus graph G, the end vertices of P be a and b and w be a vertex in P . It is clear that both Ga. Lemma 10 If B is a CEB of cactus graph G with. and Gb have at least one vertex in a g-set D of. order ≥ 4, then B contributes f − (B)−1 to f − (G). G. Let a0 ∈ Ga and b0 ∈ Gb be two vertices of D.. and f + (B) − 1 to f + (G).. Suppose to the contrary that w is in some g-set D. By definition, there is a vertex u 6∈ D such. Proof. Let v be the cut vertex of B and D be a. that u ∈ I(w, w0 ) and u 6∈ I(D \ {w}) where w0 ∈. g-set of G. It can be seen that G − B has at least. D. Since P is short, V (P ) ⊆ I(a0 , b0 ) and hence.

(6) u 6∈ V (P ). There are three cases to be considered. to b n2 c. At first, we prove that if |P | = n − 1 and n. depending on the position of u.. is odd, then vx belongs to every g-set of G. Since n is odd, b n2 c =. Case 1: u ∈ Ga . Since u ∈ I(w, w0 ), w0 must be in Ga . Therefore, 0. 0. x≤. 0. u lies on a a − w geodesic P . However, P is indeed a subpath of a b0 − w0 geodesic. Thus, u 0. n−1 2 .. Then,. n−1 2. (1). and. 0. also lies on a b − w geodesic. This contradicts |P | − x ≤. that u 6∈ I(D \ {w}). Case 2: u ∈ Gb .. n−1 . 2. (2). According to Equations 1 and 2, we have x =. This proof is similar to Case 1.. n−1 2 .. Thus, vx is the exactly one vertex that belongs to. Case 3: u ∈ B − P .. every g-set.. Since u ∈ I(w, w0 ), w0 must be in B − P . Let P 0 0. 0. Now, we prove that if P has exactly one vertex. be a w − w geodesic where u ∈ V (P ). Clearly,. vx in every g-set of G, then |P | = n − 1 and n is. a − w or b − w geodesic is a subpath of P 0 . If a − w. odd. Since, by vertex symmetry property of cycle. 0. 0. 0. geodesic is a subpath of P , then u ∈ I(a , w ), a. B, v n−1 is the only possible vertex of B in D, n−1. contradiction. Otherwise, b − w geodesic must be. is a multiple of 2. Therefore, n is odd. Suppose. a subpath of P 0 . Therefore, u ∈ I(b0 , w0 ). It is. to the contrary that |P | ≤ n − 2. Since. also a contradiction.. less than. 2. We conclude that w cannot be a vertex of any. n 2,. |P | 2. + 1 is. D \ {v |P | } ∪ {v |P | +1 } is also a g-set 2. 2. of G. This contradicts the fact that v |P | is the 2. g-set.. exactly one vertex in V (P ) ∩ D. We conclude that Q. E. D.. |P | = n − 1 and n is odd. Q. E. D.. Lemma 12 Let B be a CIB with order n of cactus graph G and P be a long segment of B. Then, P has exactly one vertex in every g-set of G if and only if |P | = n − 1 and n is odd. Proof.. Let a0 ∈ Ga and b0 ∈ Gb be. P : a = v1 , v2 , . . . , v|P | = b is long, V (P ) 6⊆ 0. I(a , b ).. the forcing numbers of G are determined by Lemmas 2 and 9. For convenience, we use α and β to denote the cardinalities of the CEBs of order 5. two vertices of a g-set D of G. Since segment. 0. Consider a cactus graph G, if G is a cycle, then. Thus, there is at least one vertex in. and order ≥ 7, respectively. Theorem 13 If G is a cactus graph and not a cycle, then f − (G) = α + β + l(G) and f + (G) = α + 2 · β + l(G).. V (P )∩D such that V (P ) ⊆ I(D). Moreover, since V (P ) ⊆ I(a0 , vb |P | c )∪I(b0 , vb |P | c ), |V (P )∩D| = 1.. Proof. If v ∈ V (G) does not belong to a CEB or. Let vx be the vertex in V (P ) ∩ D.. Then, x. a CIB of G, then v is clearly either an end vertex. and |P | − x are length of path:v1 , v2 , . . . , vx and. or a cut vertex. By Lemma 4 and Corollary 5, v. path:vx , vx+1 , vx+2 , . . . , vn−1 , respectively.. Note. cannot be a forcing vertex. Moreover, every even. that both x and |P | − x must be less than or equal. CEB contains no forcing vertex due to Lemmas 2,. 2. 2.

(7) 9 and 10. Therefore, we next consider odd CEBs. 4. Concluding Remarks. of order ≥ 5. By Lemmas 9 and 10, each odd CEB of order ≥ 5 contributes 1 to f − (G), while other CEBs have no contribution. Therefore, all CEBs totally contribute α + β to f − (G). Similarly, by Lemmas 2 and 10, each odd CEB of order 5 con-. Zhang determined the upper forcing geodetic numbers for trees, cycles, complete bipartite graphs and hypercubes [12]. In contrast, we further propose another graph parameter, namely. tributes 1 to f + (G) and each odd CEB of order. lower forcing geodetic number, and explore the. ≥ 7 contributes 2 to f + (G). Thus, all CEBs to-. lower and upper forcing geodetic numbers of. tally contribute α + 2 · β to f + (G).. block-cactus graphs. An obvious continuation of this work is to investigate the forcing geodetic. Now we are at a position to consider the con-. numbers on other larger classes of graphs. An-. tribution of CIBs. Let P be a segment in CIB B. other line of progression will be to develop efficient. of cactus graph G and the end vertices of P be a. algorithms for finding the set of forcing geodetic. and b. Let D be a g-set of G. If P is short or. vertices.. |P | = n − 1 and n is odd, then, by Lemmas 11 and 12, P has no contribution to f (D). We then check. References. LCIBs. By definition and Lemma 12, every LCIB contains exactly one vertex w in D, and w does. [1] F. Buckley and F. Harary,. Geodetic. not belong to all g-sets of G. That is, each LCIB. games for graphs, Quaestiones Mathemati-. contributes 1 to f (D). Thus, all CIBs totally con-. cae, Vol.8, No.4, pp.321–334, 1986.. tribute l(G) to f (D). Q. E. D.. [2] F. Buckley, F. Harary, and L.V. Quintas, Extremal results on the geodetic number of. For a block-cactus graph G, if v ∈ V (G) does not belong to a cyclic block, then v must be an extreme vertex or a cut vertex. By Lemma 4 and Corollary 5, v cannot be a forcing vertex. Thus, we immediately sum up the contribution of all cyclic blocks for finding f − (G) and f + (G). Since the proof is similar to the proof of Theorem 13, we. a graph, SCIENTIA Series A: Mathematical Sciences, Vol. 2, pp.17–26, 1988. [3] G. Chartrand, H. Gavlas, F. Harary and R.C. Vandell, The forcing domination number of a graph, Journal of Combinatorial Mathematics and Combinatorial Computing, Vol.25, pp.161–174, 1997.. conclude the result as follows. [4] G. Chartrand, F. Harary and Ping Zhang, Extremal problems in geodetic graph theory, Congressus Numerantium, Vol.130, pp.157– Theorem 14 If G is a block-cactus graph and not a cycle, then f − (G) = α + β + l(G) and f + (G) = α + 2 · β + l(G).. 168, 1998. [5] G. Chartrand and Ping Zhang, Realizable ratios in graph theory: geodesic parameters,.

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