Abstract

For a connected graph J, a subset is termed as a locating-total dominating set if for , and for . The number of elements in a smallest such subset is termed as the locating-total domination number of J. In this paper, the locating-total domination number of unicyclic graphs and bicyclic graphs are studied and their bounds are presented. Then, by using these bounds, an upper bound for cacti graphs in terms of their order and number of cycles is estimated. Moreover, the exact values of this domination variant for some families of cacti graphs including tadpole graphs and rooted products are also determined.

1. Introduction

Let be a simple undirected and connected graph with vertex set and edge set . The number of vertices in J is called the order of J. The set of vertices in adjacent to any vertex is called the open neighbourhood of in , denoted by , and the set is called the closed neighbourhood of in , denoted by . A vertex of degree one is called leaf (pendant) and its unique neighbouring vertex is called support vertex. A set  ⊆  (J) is said to be a dominating set of J if every vertex in  −  has nonempty open neighbourhood in and furthermore if for every vertex , then this set is referred to as a total dominating set in J. A total dominating set with the additional condition that every pair of distinct vertices outside have distinct open neighbourhood in is known as a locating-total dominating set, abbreviated as LTD-set. The smallest dominating, total dominating, and locating-total dominating sets of J are denoted by γ (J)-set,-set, and -set, respectively, and their cardinalities are known as domination, total domination, and locating-total domination numbers, respectively.

A path and cycle having m vertices are denoted by Pm and Cm. A simple connected graph having no cycle is a tree. A graph is a unicyclic or bicyclic graph if it contains exactly one cycle or two cycles, respectively. A graph in which any two distinct cycles have maximum one vertex in common is termed as a cactus. A tadpole graph is obtained by joining a leaf (pendant) vertex of path to a vertex of a cycle through an edge. For connected graphs K and L, the rooted product of K by L is obtained by considering |K| copies of L and identifying the i-th vertex of L with the root vertex of i-th copy of L. The idea of locating-total domination and differentiating-total domination was presented first by Haynes et al. [1]. They studied these variants of domination for trees and estimated a bound on this class. The differentiating-total and locating-total domination numbers of path graphs are also determined. The improved new bound for locating-total dominating number of trees is presented by Chen and Sohn [2]. They also gave the sharpness of these bounds and identified the trees achieving these bounds. The different versions of locating-total domination number for different classes of graphs are investigated in [3–5]. The bounds on locating-total domination number for different types of products are studied in [6, 7]. A new bound of locating-total domination number using annihilation number of a tree was presented by Ning et al. [8]. They also identified the trees attaining this bound. In this paper, locating-total domination number of unicyclic and bicyclic graphs is investigated and their upper bounds are given. Consequently, using these bounds, an upper bound for cacti graphs is presented. Moreover, the exact values of this domination variant for some classes of cacti graphs are computed. In particular tadpole graphs, rooted products of path by cycle graph and cycle by path graph are computed.

2. Bounds of Locating-Total Domination Number of Cacti Graphs

Theorem 1 (see [2]). For a tree having r ≥ 3 vertices and pendant vertices, one has

Proposition 1 (see [9]). The -set of a graph J must include all the support vertices of J.
A unicycle graph U having r ≥ 3 vertices can be obtained by connecting any two nonadjacent vertices of a tree having r ≥ 3 vertices by an edge.

Lemma 1. Let be a -set of a tree with order r ≥ 4 and pendant vertices. Let U be the unicyclic graph of order r ≥ 4 with pendant vertices constructed by joining any two nonadjacent vertices through an edge. If either both belong to or none of them belongs to , then

Proof. It is easy to see that the -set of a graph obtained by adding edges between two vertices of -set or between two vertices not in -set of a graph is the same as that of -set of . This implies that also becomes a -set for U. Hence, by Theorem 1, we haveTwo cases arise: Case 1: Assume that one of them is a pendant vertex. If exactly one is a pendant vertex, then and by (3), But if both are leaves, then and by (3), Case 2: Now assume none of them is a leaf; then, and by (3),

Theorem 2. Let U be a unicycle graph having r ≥ 3 vertices and pendant vertices. Then,

Proof. Let be a tree of order r with l’ leaves and be its two nonadjacent vertices such that unicycle U is obtained from by joining u and . Let be a -set of . Now the vertices and may belong to or not. If both vertices u and belong to or both vertices and do not belong to , then by Lemma 1, it is easy to see that becomes -set for U andNow suppose , and is a pendant vertex. Then there exists support vertex s such that . By Proposition 1, s ∈ . Now as is a pendant, in , and is not adjacent to any vertex of Also, u is adjacent to some vertex so that in U, As in U, is not adjacent to any other vertex of except u, so for all Moreover, for all because for all and remain the same in both and U. This further implies that, in this case, again becomes -set for U. Therefore, by Theorem 1 and the fact that either or we haveFinally, suppose that all and is not a leaf. Then, is LTD-set for U. Therefore, by Theorem 1, we haveIf u is a pendant vertex, then and by (10), we haveBut if u is not a pendant vertex, then and by (10)The result follows from (8), (9), (11), and (12).
If in a unicycle graph U with cycle length m, all the edges of its unique cycle are removed, then resulting graph is a disconnected graph which is union of m disjoint subtrees , where Each subtree is a maximal subtree of U containing only one cycle vertex. A bicyclic graph B can be constructed from a noncorona unicyclic graph U by choosing any two nonadjacent vertices for some i (i.e., and must be in one subtree of U) and joining the vertices must contain at least one cycle edge, and through an edge. It is easy to see that any path between vertices of two distinct subtrees.

Lemma 2. Consider a unicyclic graph U with r > 5 vertices and leaves and -set . Choose any two nonadjacent vertices u and from maximal subtree Ti of U containing exactly one cycle vertex such that either both belong to or none of them belongs to Let B be a bicyclic graph having r > 5 vertices and l pendant vertices obtained by connecting through an edge. Then,

Proof. The proof is the same as that of Lemma 1.

Theorem 3. For bicyclic graph B with order r > 4 and l pendant vertices, one has

Proof. Let U be the unicyclic graph of order r > 4 with pendant vertices and be its two nonadjacent vertices, where is maximal subtree of U containing exactly one cycle vertex. Then, bicyclic graph B can be obtained from U by joining u and . Let be a -set of U. Now the vertices u and may belong to or not.
The result follows by using Lemma 2 and the similar arguments as used in Theorem 2.
Now, as an extension of Theorems 2 and 3, an upper bound for locating-total domination number of cacti graphs is appraised as follows.

Theorem 4. Let be cacti with r vertices, l pendant vertices, and q cycles. Then, one has

Proof. We prove this by making use of mathematical induction on q. If q = 1, 2, then the desired result follows immediately by Theorems 2 and 3. Hence, we have a base for induction. Now assume that the bound is satisfied for any cacti graph having order r and containing q − 1 cycles. Now a cactus with order r, leaves l, and cycles q can be obtained from a cactus of order r with leaves, q − 1 cycles, and -set in one of the following three ways: Select any two pendant vertices u and of such that the shortest path between u and must not have any cycle edge. Now if the vertices u and are connected by an edge, we get a cactus . In this case, we can see that the set is also LTD-set of by using similar arguments as used in Theorems 2 and 3. Now here and using induction hypothesis, we have Select a pendant vertex u and a nonpendant vertex of such that the shortest path between u and does not include any cycle edge. Now if the vertices u and are connected by an edge, we get a cactus . In the case, when is LTD-set of ; otherwise, is LTD-set for by using similar arguments as used in Theorems 2 and 3. Now here and using induction hypothesis, we have Select any two nonpendant vertices u and of such that the path between u and does not include any cycle edge. Now if the vertices u and are connected by an edge, we get a cactus . In the case, when exactly one of the u or , say u, belongs to , then is LTD-set; otherwise, is LTD-set for by using similar arguments as used in Theorems 2 and 3. Now here and using induction hypothesis, we haveHence, the result follows.

3. Tadpole and Rooted Product Graphs

In this section, locating-total domination number for few cacti graphs is determined. In particular, tadpole graphs (unicyclic graphs), rooted product of cycle by path (unicyclic graphs), and rooted product of by (cacti graph with q cycles) are investigated.

Theorem 5 (see [1]). For

Theorem 6 (see [4]). For

Remark 1. The locating-total domination number of and for different values of q can be simplified as(1) if .(2) if .(3) if q is odd.Let be a tadpole having vertices and is formed by connecting a vertex of cycle to pendant vertex of path by an edge.

Theorem 7. Let be odd positive integer. Then, one has

Proof. Suppose that : a1 ⟶ a2 ⟶ a3 ⟶  ···  ⟶ aq2 is connected through to a cycle vertex u of . Let be a -set for . As || is minimum, therefore a2 being support vertex by Proposition 1 and a3 being its unique neighbouring vertex by definition of locating-total dominating set must belong to . Consequently, a4 and a5 can be excluded from and then the next two consecutive vertices belong to . By continuing this process, we can see that for and for . Case 1: Suppose q2 ≡ 1 (mod4). If q2 = 1, then is a pendant vertex and u is a support vertex. Thus, and But if q2 > 1, then as q2 ≡ 1 (mod4), and  ∈ . Consequently, total vertices of belong to . Since , , therefore u ∈ . Thus, we have The -set of tadpole graph for q ≡ 1 (mod4) is represented by black vertices in Figure 1. Case 2: Suppose q2 ≡ 2 (mod4). If q2 = 2, then is a support vertex. Therefore,  ∈ . Also, using the definition of LTD-set, u ∈ . This implies that cycle vertices must belong to and hence we have But if q2 > 2, then as q2 ≡ 2 (mod4), therefore  ∈ D and , . Consequently, path vertices belong to . Also, since  ∈  and , , therefore u ∈  which implies that cycle vertices must be in . Hence, The -set of tadpole graph for q ≡ 2 (mod4) is represented by black vertices as shown in Figure 2. Case 3: Now assume q2 ≡ 0 (mod4). If q2 = 0, then we have But if q2 > 0, then as q ≡ 0 (mod4) and || is minimum, so and , ∈ . Consequently, path vertices lie in . Also, since and , ∈ and u can be excluded from . This implies that cycle vertices must lie in and hence The -set of tadpole for is shown in Figure 3 with black vertices. Case 4: Finally, assume that q2 ≡ 3 (mod4). If q2 = 3, then a2 being support vertex and a3 being its neighbouring vertex lie in . But then u can be taken out of . Moreover, for cycle vertices lie in and henceOn the other hand, if q1 ≡ 1 (mod4), then cycle vertices belong to and we haveNow if q2 > 3, then as q2 ≡ 3 (mod4); therefore, ,  ∈  and . Consequently, path vertices lie in . Also, as ,  ∈ , and u can be excluded from . But then the vertices which lie in must lie in . Moreover, if q1 ≡ 3 (mod4), then cycle vertices are in . Thus, we havewhereas if q1 ≡ 1 (mod4), then cycle vertices belong to and we haveThe -set of tadpole for is shown in Figure 4 with black vertices.


Figure 1 
Tadpole for q2 ≡ 1 (mod4).

Figure 2 
Tadpole for q ≡ 2 (mod4).

Figure 3 
Tadpole for q2 ≡ 0 (mod4).

Figure 4 
Tadpole for .

Theorem 8. Let q1 be an even integer. Then one has

Proof. Let be a -set for . As |W| is minimum, therefore a2 being support vertex by Proposition 1 and a3 being its unique neighbouring vertex by definition of locating-total dominating set must belong to . Consequently, a4 and a5 can be excluded from and then the next two consecutive vertices belong to . By continuing this process, we can see that for j ≡ 2, 3 (mod4), and for j ≡ 0, 1 (mod4), . Case 1: Suppose q2 ≡ 1 (mod4). If q2 = 1, then is a pendant vertex and u is a support vertex. Thus,  ∈  and The -sets of tadpoles and are shown in Figure 5 with black vertices. But if q2 > 1. Then as q2 ≡ 1 (mod4), W and  ∈ . Consequently, total path vertices belong to . Further, since , therefore u ∈ . Thus, cycle vertices lie in D and consequently, Case 2: Suppose q2 ≡ 3 (mod4). If q2 = 3, then a2 being support vertex and a3 being its neighbouring vertex lie in . And u can be excluded from . This implies that cycle vertices must lie in , and hence, we have The -set of tadpoles and are shown in Figure 6 with black vertices. But if q2 > 3, then as q2 ≡ 3 (mod4), so lie in , whereas . Consequently, path vertices belong to . Further, as lie in so u can be taken out of . Thus, cycle vertices lie in so that Case 3: Now suppose q2 ≡ 2 (mod4). If q2 = 2, then being a support vertex belongs to . This further implies that u ∈ . Hence, cycle vertices lie in , and consequently, The -set of tadpoles and are shown in Figure 7 with black vertices. On the other hand, if q2 > 2, then aq2 ∈  but , . This implies that path vertices are included in along with cycle vertex u. Consequently, cycle vertices lie in and we have Case 4: Finally assume q2 ≡ 0 (mod4). It is easy to see that if q2 = 0, then and r = q1.Hence,Using Remark 3, we haveAs r = q1, so we haveBut if q2 > 0, then and ,  ∈ . This shows that path vertices belong to . Also, since and ,  ∈ , u must be in . Hence, the vertices that belong to must lie in . Hence, we haveAgain, using Remark 3, we haveThus,In Theorem 9, the rooted product is computed. Any cycle vertex can be considered as root vertex.

(a)
(a)
(b)
(b)
Figure 5 
The tadpoles and .
(a)
(a)
(b)
(b)
Figure 6 
The tadpoles and .
(a)
(a)
(b)
(b)
Figure 7 
The tadpoles and .

Theorem 9. For positive integers q1 ≥ 3 and q2 ≥ 1, one has

Proof. If q2 = 1, then and Now we may suppose q2 ≥ 2.
Then four cases arise: Case 1: Suppose , that is, for integer k > 0. The graph contains mutually disjoint cycles , where i = 1,·..., . For each i, cycle consists of 4k vertices, namely, where j = 1, …, 4k. Take as root vertex of each cycle. The 4k vertices in each cycle are divided into k sets such that each set contains four consecutive vertices of cycle. From each set, the middle two vertices can be taken as elements of minimum LTD-set for each . Let this minimum LTD-set of Then with | = 2k =  These 2k vertices in each are shown in Figure 8 as black vertices. Now the root vertices from each are actually the path vertices and are locating totally dominated by the elements of for each i. Hence, all the vertices of are locally dominated by Also, this set is minimum by construction. Thus, Case 2: Suppose , that is, for integer k > 0. Let be the root vertex of cycle for each i. The graph contains 4k vertices and these 4k vertices are divided into k sets. Now we select two middle vertices from each set. Then, all 4k vertices of are locating totally dominated by 2kchosen vertices. Since does not have a neighbour in the selected LTD-set of , therefore, must be a LTD-vertex. The graph also contains 4k vertices and these 4k vertices are again divided into k sets and all are locating totally dominated by 2k vertices by selecting the first and last vertices from each set. Now we are interested in minimum LTD-set; may not be an LTD-vertex. By continuing in this way, we can see that if i is odd then is not an LTD-vertex otherwise is an LTD-vertex. This along with the fact that path vertices are root vertices implies that all the vertices of graph are locating totally dominated by a set of smallest cardinality. We note that if the path (root) vertex is not an LTD-vertex, then associated cycle has 2k vertices in and otherwise each cycle has 2k + 1 vertices in . Hence, The -set of for q1 ≡ 1 (mod4) is shown in Figure 9 with black vertices. Case 3: Suppose q1 ≡ 2 (mod4) such that q1 = 4k + 2 and k > 0 is an integer. In a similar manner as in Cases 1 and 2, we divide the vertices of each where ai is neighbouring vertex of root vertex into k sets. For each i, all the vertices of are locating totally dominated by 2k chosen vertices (middle two vertices in each set). The vertices and ai for all i can be locating totally dominated if we include all the root vertices. This implies that The -set of for q1 ≡ 2 (mod4) is shown in Figure 10 with black vertices. Case 4: Finally, suppose q1 = 4k + 3, where k > 0 is an integer. Here, the vertices of where ai and bi are consecutive two vertices of root vertex on cycle, are divided into k sets. Again for each i, all the vertices are locating totally dominated by 2k chosen vertices (first two vertices in each set). The vertices for all i can be locating-total dominated by selecting two of them except root vertex. Therefore, in this case, we haveThe -set of for q1 ≡ 3 (mod4) is shown in Figure 11 with black vertices.
In the following Theorem 10, the rooted product is computed. Here, we select pendant vertex of path as root vertex.


Figure 8 
The rooted product for q1 ≡ 0 (mod4).

Figure 9 
The rooted graph for q1 ≡ 1 (mod4).

Figure 10 
The graph for q1 ≡ 2 (mod4).

Figure 11 
The graph for q1 ≡ 3 (mod4).

Figure 12 
The product for q2 ≡ 0 (mod4).

Figure 13 
The graph for q2 ≡ 1 (mod4).

Figure 14 
The product for q2 ≡ 2 (mod4).

Theorem 10. For q1 ≥ 3 and q2 ≥ 2 and by taking pendant vertex of as root vertex, one has

Proof.  Case 1: Suppose q2 ≡ 0 (mod4), such that q2 = 4k and k > 0. The rooted product consists of mutually disjoint q1 paths for i = 1, ..., q1. Suppose is the root vertex of for i = 1, ..., q1. The 4k vertices of are split into k sets, each containing four consecutive vertices. Then the collection of middle two vertices from each set in every path gives the LTD-set for graph which is also minimum one: The -set of for q2 ≡ 0 (mod4) is represented by black vertices in Figure 12. Case 2: Assume q2 ≡ 1 (mod4) such that q2 = 4k + 1 and k > 0. Let be the root vertex in each path of Consider consisting of 4k vertices and all the vertices in can be locating totally dominated by 2k chosen vertices (two middle vertices from each set). Since all the paths for i = 1,···, q1 are mutually disjoint, therefore is minimum locating-total dominated set consisting of vertices. Therefore, The -set of for q2 ≡ 1 (mod4) is represented by black vertices as shown in Figure 13. Case 3: Now suppose that q2 ≡ 2 (mod4) such that q2 = 4k + 2. Here, each for i = 1,···, q1 consists of 4k + 2 vertices. All of them except the root and its neighbouring vertex are locating totally dominated by 2k chosen vertices from each path as in the previous cases. These chosen vertices must belong to -set. To obtain, minimum LTD-se for rooted product, all the root vertices are also included. Hence, The -set of for q2 ≡ 2 (mod4) is represented by black vertices in Figure 14. Case 4: The case q2 ≡ 3 (mod4) is dealt in a similar manner as in previous cases. Thus, we have

The -set of for q2 ≡ 3 (mod4) is shown in Figure 15 with black vertices.


Figure 15 
The graph for q2 ≡ 3 (mod4).

4. Conclusion

In this paper, the upper bounds for locating-total domination number of unicyclic and bicyclic graphs are obtained. It is shown that, for unicyclic graph U of order r and leaves l, and for bicyclic graph B of order r and leaves l, These bounds are then generalized for cacti graphs of order n having q cycles as . Also, the exact value of locating-total domination numbers for some cacti graphs is determined. For tadpole graph , it is found that locating-total dominating number is , , or depending upon the parity of r and values of q1 and q2. The locating-total dominating number of rooted product of Cq1 by Pq2 in terms of q1 and is computed. Similarly, locating-total dominating number of rooted products of Pq2 by Cq1 in terms of q2 and is also determined. It is still an open problem to find the sharpness of estimated bounds of locating-total dominating number for cacti graphs.

Data Availability

All the required data are included within the paper. However, the reader may contact the corresponding author for more details of the data.

Conflicts of Interest

The authors do not have any conflicts of interest.

Acknowledgments

This work was sponsored by Shandong Provincial Natural Science Foundation, China (ZR2018MA010).