On Locating-Dominating Set of Regular Graphs

<jats:p>Let <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1">
                        <mi>G</mi>
                     </math>
                  </jats:inline-formula> be a simple, connected, and finite graph. For every vertex <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M2">
                        <mi>v</mi>
                        <mo>∈</mo>
                        <mi>V</mi>
                        <mfenced open="(" close=")" separators="|">
                           <mrow>
                              <mi>G</mi>
                           </mrow>
                        </mfenced>
                     </math>
                  </jats:inline-formula>, we denote by <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M3">
                        <msub>
                           <mrow>
                              <mi>N</mi>
                           </mrow>
                           <mrow>
                              <mi>G</mi>
                           </mrow>
                        </msub>
                        <mfenced open="(" close=")" separators="|">
                           <mrow>
                              <mi>v</mi>
                           </mrow>
                        </mfenced>
                     </math>
                  </jats:inline-formula> the set of neighbours of <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M4">
                        <mi>v</mi>
                     </math>
                  </jats:inline-formula> in <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M5">
                        <mi>G</mi>
                     </math>
                  </jats:inline-formula>. The locating-dominating number of a graph <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M6">
                        <mi>G</mi>
                     </math>
                  </jats:inline-formula> is defined as the minimum cardinality of <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M7">
                        <mi>W</mi>
                        <mtext> </mtext>
                        <mo>⊆</mo>
                        <mtext> </mtext>
                        <mi>V</mi>
                        <mfenced open="(" close=")" separators="|">
                           <mrow>
                              <mi>G</mi>
                           </mrow>
                        </mfenced>
                     </math>
                  </jats:inline-formula> such that every two distinct vertices <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M8">
                        <mi>u</mi>
                        <mo>,</mo>
                        <mi>v</mi>
                        <mo>∈</mo>
                        <mi>V</mi>
                        <mfenced open="(" close=")" separators="|">
                           <mrow>
                              <mi>G</mi>
                           </mrow>
                        </mfenced>
                        <mo>\</mo>
                        <mi>W</mi>
                     </math>
                  </jats:inline-formula> satisfies <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M9">
                        <mo>∅</mo>
                        <mo>≠</mo>
                        <msub>
                           <mrow>
                              <mi>N</mi>
                           </mrow>
                           <mrow>
                              <mi>G</mi>
                           </mrow>
                        </msub>
                        <mfenced open="(" close=")" separators="|">
                           <mrow>
                              <mi>u</mi>
                           </mrow>
                        </mfenced>
                        <mo>∩</mo>
                        <mi>W</mi>
                        <mo>≠</mo>
                        <msub>
                           <mrow>
                              <mi>N</mi>
                           </mrow>
                           <mrow>
                              <mi>G</mi>
                           </mrow>
                        </msub>
                        <mfenced open="(" close=")" separators="|">
                           <mrow>
                              <mi>v</mi>
                           </mrow>
                        </mfenced>
                        <mo>∩</mo>
                        <mi>W</mi>
                        <mo>≠</mo>
                        <mo>∅</mo>
                     </math>
                  </jats:inline-formula>. A graph <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M10">
                        <mi>G</mi>
                     </math>
                  </jats:inline-formula> is called <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M11">
                        <mi>k</mi>
                     </math>
                  </jats:inline-formula>-regular graph if every vertex of <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M12">
                        <mi>G</mi>
                     </math>
                  </jats:inline-formula> is adjacent to <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M13">
                        <mi>k</mi>
                     </math>
                  </jats:inline-formula> other vertices of <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M14">
                        <mi>G</mi>
                     </math>
                  </jats:inline-formula>. In this paper, we determine the locating-dominating number of <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M15">
                        <mi>k</mi>
                     </math>
                  </jats:inline-formula>-regular graph of order <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M16">
                        <mi>n</mi>
                     </math>
                  </jats:inline-formula>, where <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M17">
                        <mi>k</mi>
                        <mo>=</mo>
                        <mi>n</mi>
                        <mo>−</mo>
                        <mn>2</mn>
                     </math>
                  </jats:inline-formula> or <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M18">
                        <mi>k</mi>
                        <mo>=</mo>
                        <mi>n</mi>
                        <mo>−</mo>
                        <mn>3</mn>
                     </math>
                  </jats:inline-formula>.</jats:p>


Introduction
Given a simple, connected, and finite graph G. e neighbours of a vertex v of G are defined as the vertex set N G (v) � u ∈ V(G)|uv ∈ E(G) { }. A set of vertices W ⊆ V(G) is called a locating-dominating set of a graph G if every two distinct vertices u, v ∈ V(G)\W satisfies ∅ ≠ N G (u) ∩ W ≠ N G (v) ∩ W ≠ ∅. e minimum cardinality of locatingdominating sets of G is called the locating-dominating number of G, denoted by λ(G). is concept was introduced by Slater [1,2].
Let us model a building floor as a graph. Locatingdominating set can be used to determine an exact location of a fire alarm which sends a signal when detecting a fire in any of its adjacent vertices. e activated signals will univocally determine the place of the fire.
Furthermore, some authors have been characterized all graphs with a given locating-dominating number. Henning and Oellermann [6] have proved that for a connected graph G of order n ≥ 2, λ(G) � n − 1 if and only if G is either complete graphs K n or star graphs K 1,n−1 . ey also characterized all graphs G of order n ≥ 4 with locating-dominating number n − 2. Meanwhile, Caceres et al. [7] have proved that there are 16 nonisomorphic graphs G satisfying λ(G) � 2.
Some authors also have determined the locating-dominating number of graphs obtained from a product graphs. Canoy and Malacas [14] provided the lower and upper bounds for the locating-dominating number of corona product graphs. ey also determined an exact value of the locating-dominating number of a composition of graphs between G and H where G is a connected totally point determining graph and H is a nontrivial connected graph. An exact value of the locating-dominating number of comb product of any two connected graphs of order at least two has been determined by Pribadi and Saputro [15]. Murtaza et al. [16] studied the locating-domination number of functigraphs of complete graphs. A study of a locatingdominating set of a graph by adding a universal vertex can be seen in [17].
In this paper, we consider a regular graph. A graph G is called k-regular graph if every vertex in G is adjacent to k other vertices. Since every vertex of G is adjacent to the same number of vertices of G, every vertex of G has the same probability to distinguish some distinct vertices of G. Let G � (V, E) be a model of the multiprocessor system, such that V(G) is the set of processors and E(G) is the set of links between processors. Assume that at most one processor is malfunctioning and we want to test the system and find the faulty processor. Some processors can be chosen and assigned to check their neighbours. In case a selected processor detects a fault, it sends an alarm signal. Since we need an exact location of a faulty processor, we must choose some processors such that the chosen processors can uniquely tell the location of the malfunctioning processor. en, a locating-dominating set of G can be used to choose those processors.
Bertrand et al. [4] have initiated the study of the locatingdominating number on regular graph. ey determined the locating-dominating number of 2-regular connected graphs (cycles). e locating-dominating number of (n − 1)-regular graph of order n can be seen in [7]. In this paper, we determine the locating-dominating number of k-regular graph of order n where k � n − 2 or k � n − 3. e purpose of this paper is to further investigate the locating-dominating number of certain family of graphs, namely, to determine the locating-dominating number of certain regular graphs. We obtain two main results, one of them is the following result related with an (n − 2)-regular graph. Our second result is related with an (n − 3)-regular graph. In preparing the proof for the second result, we are able to obtain the intermediate result as follows.
For n ≥ 5, we consider certain cycles contained in a complete graph K n . In this paper, we assume that a cycle contains at least three vertices. For r ∈ 1, 2, . . . , ⌊n/3⌋ { }, let R 1 , R 2 , . . . , R r be r disjoint cycles contained in K n such that Note that an (n − 3)-regular graph is isomorphic to K n (E(R 1 ) ∪ E(R 2 ) ∪ · · · ∪ E(R r )). In case r � 1, the locating-dominating number of an (n − 3)-regular graph of order at least 7 has been determined in eorem 2. In eorem 3, we provide the locating-dominating number of an (n − 3)-regular graph of order n ≥ 5 with 1 ≤ r ≤ ⌊n/3⌋. Theorem 3. For n ≥ 5 and 1 ≤ r ≤ ⌊n/3⌋, let R 1 , R 2 , . . . , R r be r disjoint cycles contained in K n such that If k is the number of cycles R i of order m i ≥ 7 and m i ≡ 1 or 3 (mod 5), then 2. (n − 2)-Regular Graph and Proof of Theorem 1 eorem 1 is a direct consequence of Lemmas 1 and 2 in this section.
In this section, we define G as an (n − 2)-regular graph of order n ≥ 4. Note that if we count the sum of degree of all vertices, then every edge will be counted twice. erefore, we have 2|E(G)| � n(n − 2). It implies that n must be even.
Let us consider y i and y j for 1 ≤ i < j ≤ (n/2). Since

(n − 3)-Regular Graph and Proof of Theorem 2
In this section, we define G as an (n − 3)-regular graph of order n ≥ 5. Note that G contains a subgraph which is isomorphic to en, every vertex in G ′ is adjacent to all vertices of G\G ′ . In Lemma 3, we show that every subgraph is contributed to a locating-dominating set of G, which is a direct consequence of Observation 1 which has been proved by Henning and Oellermann [6].
Observation 1 (see [6]). Let W be a locating-dominating set of a connected graph G. If there exists two distinct vertices From Lemma 3, we have Lemma 4.
can be an empty set. In Lemma 5, we provide a locating-dominating set of a subgraph e lower bounds are sharp. Proof.
For the sharpness, we define a vertex set S m as follows: We will show that S m is a locating-dominating set of a subgraph G ′ of G. Let us consider vertices in It is easy to see that We can say that the locating-dominating number of G ′ in Lemma 5 is given by Now, let us consider G ′ ⊆ G and G ′ � K m \E(C m ) for m ≥ 7. us, the order of G must be n ≥ 7. For n ≥ 7, in order to determine a locating-dominating set of G ′ � K m \E(C m ) for 7 ≤ m ≤ n, we define some definitions. ese definitions were firstly introduced by Buczkowski et al. [18]. ey used this gap technique to determine the metric dimension of wheel graphs. In lemma 6, we provide the necessary and sufficient conditions for a locating-dominating set of G ′ which is related to gap definition.

n}. e vertex set W ⊆ V(G ′ ) is a locating-dominating set of G ′ if and only if W satisfies all conditions as follows: (1) Every gap with respect to W contains at most 3 vertices (2) W contains at most one gap of 3 vertices (3) If A is a gap with respect to W, containing 2 or 3 vertices, then any neighbouring gaps of A have at most one vertex
Proof. (⇒) We will prove all three conditions by contradiction.
(1) Suppose that there exists a gap with respect to W containing at least 4 vertices.
Let a 1 , a 2 , a 3 , a 4 ∈ V(G ′ ) where a i a i+1 ∈ E(C m ) with 1 ≤ i ≤ 3 and all those vertices are not in W. en, we have (2) Suppose that there exists two gaps containing 3 vertices. Let a 1 , a 2 , a 3 and b 1 , b 2 (3) Suppose that there exists a neighbouring gap of A containing at least 2 vertices. Let a 1 , a 2 , a 3 , a 4 , a 5 ∈ V(G ′ ) where a i a i+1 ∈ E(C m ) with 1 ≤ i ≤ 4 and a 3 is the only element of W among them. en, we have N G′ (a 2 ) ∩ W � W\ a 3 � N G′ (a 4 ) ∩ W, a contradiction.
(i) u belongs to a gap containing one vertex.
Let a and b be two end points of this gap. So, u is the only vertex which is not adjacent to a and b. Since |S| ≥ 3 and for every vertex (ii) u belongs to a gap containing two vertices.
Let a and b be two end points of this gap. Without loss of generality, let au ∈ E(C m ). So, N G′ (u) otherwise, x belongs to a gap containing one vertex. From (i) above, we obtain that ∅ ≠ N G′ (u) ∩ S ≠ N G′ (x) ∩ S ≠ ∅. (iii) u belongs to a gap containing three vertices.
Let c 1 , c 2 , and c 3 be a gap of three vertices where c i c i+1 ∈ E(C m ) with 1 ≤ i ≤ 2 and a and b be end points of this gap. Let ac 1 , bc 3 ∈ E(C m ). If u � c 2 , then N G′ (u) ∩ S � S. Since u is the only vertex having this property, we obtain that for every vertex otherwise, x belongs to a gap containing one vertex. From (i) above, we obtain that ∅ ≠ N G′ (u) ∩ S ≠ N G′ (x) ∩ S ≠ ∅. □ Now, we are ready to prove eorem 2.
Since |W| � 2k and satisfies all conditions in Lemma 6, then W is a locating-dominating set of H.
Since |W| � 2k and satisfies all conditions in Lemma 6, then W is a locating-dominating set of H.
Since |W| � 2k + 1 and satisfies all conditions in Lemma 6, then W is a locating-dominating set of H.
Since |W| � 2k + 1 and satisfies all conditions in Lemma 6, then W is a locating-dominating set of H.
Since |W| � 2k + 2 and satisfies all conditions in Lemma 6, then W is a locating-dominating set of H.
(2) λ(H) ≥ ⌈(2m − 2)/5⌉: let S be a locating-dominating set of H with minimum cardinality. We consider two following cases: (a) |S| is even. Let |S| � 2k for a positive integer k. So, the number of gap of H with respect to S is 2k. Since S must be satisfy all conditions in Lemma 6, the number of gap containing at least 2 vertices is at most k. It follows that the number of vertex which is not in S is at most 3k + 1. So, we obtain that k ≥ ((m − 1)/5). erefore, (b) |S| is odd. Let |S| � 2k + 1 for a positive integer k. So, the number of gap of H with respect to S is 2k + 1. Since S must be satisfy all conditions in Lemma 6, the number of gap containing at least 2 vertices is at most k. It follows that the number of vertex which is not in S is at most 3k + 2. So, we obtain that k ≥ ((m − 3)/5). erefore, 4 Journal of Mathematics

(n − 3)-Regular Graph and Proof of Theorem 3
For n ≥ 5, we consider certain cycles contained in a complete graph K n . For 1 ≤ r ≤ ⌊n/3⌋, let R 1 , R 2 , . . . , R r be r disjoint cycles contained in K n such that V( Note that an (n − 3)-regular graph is isomorphic to K n \(E(R 1 ) ∪ E(R 2 ) ∪ · · · ∪ E(R r )). In case r � 1, the locating-dominating number of an (n − 3)-regular graph of order at least 7 has been determined in eorem 2. Now, we will determine the locating-dominating number of an (n − 3)-regular graph of order n ≥ 5 with 1 ≤ r ≤ ⌊n/3⌋.
Considering Lemma 4, a locating-dominating set of G consists of a locating-dominating set of G i with 1 ≤ i ≤ r. erefore, we obtain that Note that a locating-dominating of G also must satisfy all three conditions in Lemma 6.
{ } such that both locatingdominating sets G i and G j contain a gap of three vertices, then we must add at least one more vertex on W. So, we need to know the gap properties of a locating-dominating set of G i . (1) If m � 3, n > m � 5, or m ≡ 0, 2, 4(mod5), then there exists a locating-dominating set of G ′ where every gap contains at most two vertices. (2) If m � n � 5 or m ≡ 1, 3(mod5) with m ≠ 3, then a locating-dominating set of G ′ has a gap containing three vertices.
□ Now, we are ready to prove eorem 3.
Proof of eorem 3. e first case for λ(G) is a direct consequence of eorem 2, Lemmas 6 and 7, and Remark 1.
For the last case, let H m 1 , H m 2 , . . . , H m k be disjoint k subgraphs of G such that H m i � K m i \E(C m i ) where 1 ≤ i ≤ k, m i ≠ 3, and m i ≡ 1 or 3(mod5). Let B i be a locating-dominating set of H m i with λ(H m i ) vertices. By Lemma 7, B i has a gap containing three vertices, say a i 1 , a i 2 , and a i 3 where a i j a i j+1 ∉ E(H m i ) with 1 ≤ j ≤ 2. We define B i ′ � B i ∪ a i 2 . It is easy to see that B i ′ is a locating-dominating set of H m i which all the gaps contain at most two vertices. So, by using this property to subgraphs H m 1 , H m 2 , . . . , H m k−1 of G, eorem 2, Lemmas 6 and 7, and Remark 1, we prove the last case.

Data Availability
is research article is the theoretical study of locatingdominating set in graphs. ere is no data supporting used. All results in the manuscript can be obtained by the analytical method.

Journal of Mathematics
Conflicts of Interest e authors declare there are no conflicts of interest.

Authors' Contributions
A.K.A.G and S.W.S conceptualized and wrote the study. S.W.S revised the study. All authors have read and agreed to the published version of the manuscript.