Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 219 https://internationalpubls.com The Forcing Circular Number of a Graph 1S. Sheeja, 2,*K. Rajendran 1Research Scholar, Vels Institute of Science Technology and Advanced Studies, Chennai, Tamil Nadu, India. 2Associate Professor, Vels Institute of Science Technology and Advanced Studies, Chennai, Tamil Nadu, India 1 Email id: sheeja1304@gmail.com 2 Corresponding author: gkrajendra59@gmail.com Article History: Received: 18-04-2024 Revised: 08-06-2024 Accepted: 20-06-2024 Abstract: Let 𝑆 be a π‘π‘Ÿ-set of graph 𝐺 and let 𝐺 be a connected graph. If 𝑆 is the only π‘π‘Ÿ-set that contains 𝑇, then a subset 𝑇 βŠ† 𝑆 is referred to be a forcing subset for 𝑆. A minimum forcing subset of 𝑆 is a forcing subset for 𝑆 of minimum cardinality. The cardinality of a minimum forcing subset of 𝑆 is the forcing circular number of 𝑆, represented by the notation π‘“π‘π‘Ÿ(𝑆). π‘“π‘π‘Ÿ(𝐺) = min {π‘“π‘π‘Ÿ(𝑆)} is the forcing circular number of 𝐺, where the minimum is the sum of all minimum forcing circular-sets 𝑆 in 𝐺. For several standard graphs, the forcing circular number is identified. It is demonstrated that there exists a connected graph G such that 𝑓𝑔(𝐺) = π‘Ž and π‘“π‘π‘Ÿ(𝐺) = 𝑏 for every integer π‘Ž β‰₯ 0, and 𝑏 β‰₯ 0. Keywords: π‘π‘Ÿ-set, circular number, forcing circular number. AMS Subject Classification: 05C12. 1. Introduction and Preliminaries A graph 𝐺 = (𝑉, 𝐸) is a connected, finite graph that does not have loops or numerous edges. 𝐺 is represented by the symbols 𝑛 and π‘š, respectively, for order and size. We use [1,6] for basic terminology in graph theoretic. If 𝑒𝑣 ∈ 𝐸(𝐺), β€œthen two vertices, 𝑒 and 𝑣, are considered nearby in 𝐺. The collection of vertices next to a vertex 𝑣 in 𝐺 is called its neighbourhood, or 𝑁(𝑣). The vertex 𝑣 has a degree of 𝑑𝑒𝑔(𝑣) = |𝑁(𝑣)|. We refer to u as an end edge, u as a leaf, and v as a support vertex if 𝑒 = {𝑒, 𝑣} is an edge of a graph G with 𝑑𝑒𝑔(𝑒) = 1 and 𝑑𝑒𝑔(𝑣) > 1. The greatest degree of a graph 𝐺 is shown by βˆ†(𝐺). 𝐺[𝑆] is the representation of the subgraph that a set 𝑆 of vertices of a graph 𝐺 induces, where 𝑉 (𝐺[𝑆]) = 𝑆 and 𝐸(𝐺[𝑆]) = {𝑒𝑣 ∈ 𝐸(𝐺) ∢ 𝑒, 𝑣 ∈ 𝑆}. A vertex 𝑣 is an extreme vertex of 𝐺 if and only if 𝐺[𝑁(𝑣)] is complete. The length of the shortest path between two vertices 𝑒, 𝑣 ∈ 𝑉(𝐺) is the distance 𝑑(𝑒, 𝑣). A 𝑒 βˆ’ 𝑣 geodesic of 𝐺 is any 𝑒 βˆ’ 𝑣 path of length 𝑑(𝑒, 𝑣). If π‘₯ is a vertex of 𝑃 and π‘₯ β‰  𝑒, 𝑣, then π‘₯ is an internal vertex of a eπ‘’βˆ’ e𝑣 path 𝑃. 𝐼[e𝑒, e𝑣] is the closed interval consisting of 𝑒, 𝑣 and all vertices that are on a eπ‘’βˆ’ e𝑣 geodesic of 𝐺. The closure of a non-empty set 𝑆 βŠ† 𝑉 (𝐺) is given by the set 𝐼[𝑆] = ⋃ 𝐼[𝑒, 𝑣]𝑒,π‘£βˆˆπ‘† . If 𝐼[𝑆] = 𝑉 (𝐺), then a set 𝑆 βŠ† 𝑉e(𝐺) is a geodetic set. The geodetic number of 𝐺, represented by 𝑔(𝐺), is the lowest cardinality of a geodetic set of 𝐺. A 𝑔 βˆ’set of 𝐺 is a geodetic set” of minimum cardinality. See [3,4,8] for references on geodetic parameters in graphs. The longest path between two vertices 𝑒, 𝑣 ∈ 𝑉(𝐺) is the detour distance 𝐷(𝑒, 𝑣). A 𝑒 βˆ’ 𝑣 detour of 𝐺 is any aπ‘’βˆ’ a𝑣 path of length 𝐷(a𝑒, a𝑣). All vertices of the closed interval 𝐼𝐷[𝑒, 𝑣] lie on some 𝑒 βˆ’ 𝑣 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 220 https://internationalpubls.com detour of 𝐺, and the interval itself consists of 𝑒, 𝑣. The closure of a non-empty set 𝑆 βŠ† 𝑉 (𝐺) is given by the set 𝐼𝐷[𝑆] = ⋃ 𝐼𝐷[𝑒, 𝑣]𝑒,π‘£βˆˆπ‘† . A detour set is then defined as a set 𝑆 βŠ† 𝑉 (𝐺). The detour number of G, represented by 𝑑𝑛(𝐺), is the lowest cardinality of a detour set of 𝐺. A 𝑑𝑛-set of 𝐺 is a diversion set with minimum cardinality. Hence [5,7] covered the study of these ideas. 𝐷𝑐(𝑒, 𝑣) represents the circular distance between 𝑒 and 𝑣, which is represented as 𝐷𝑐(a𝑒, a𝑣) = { 𝐷(a𝑒, a𝑣) + 𝑑(𝑒, 𝑣) if a𝑒 β‰  a𝑣 0 if a𝑒 = a𝑣 The detour distance and the distance between 𝑒 and 𝑣 are denoted by 𝐷(a𝑒, 𝑣) and 𝑑(a𝑒, 𝑣), respectively. The circular diameter 𝐷𝑐 is the longest circular distance between 2 vertices on 𝐺. An 𝑒 βˆ’ 𝑣 circular of 𝐺 is any 𝑒 βˆ’ 𝑣 path of length 𝐷𝑐(𝑒, 𝑣). The circular diameter 𝐷𝑐 is the longest circular distance between 2 vertices on 𝐺. For 𝑒, 𝑣 ∈ 𝑉, 𝐼𝑐[𝑒, 𝑣] represents group of every vertex positioned on a 𝑒 βˆ’ 𝑣 circular in 𝐺. For 𝑆 βŠ† 𝑉(𝐺), let 𝐼𝑐[𝑆] = ⋃ 𝐼𝑐[𝑒, 𝑣].𝑒,𝑣 βˆˆπ‘† These concepts were studied in [2,9,10]. Theorem 1.1. [2] In a connected graph, every geodetic set of 𝐺 has an extreme vertex. Theorem 1.2. [2] Let π‘Š be the set of all geodetic sets in graph 𝐺. Then π‘“π’ˆ(𝐺) ≀ 𝑔(a𝐺)– |aπ‘Š|. 2. The forcing circular number of a graph Definition 2.1. A subset 𝑇 βŠ† 𝑆 is referred to as a forcing subset for e𝑆, if 𝑆 is the only π‘π‘Ÿ-set that contains 𝑇. A forcing subset of minimum cardinality for e𝑆 is known as a minimum forcing subset of 𝑆. The forcing circular number of 𝐺 is denoted by the notation π‘“π‘π‘Ÿ(𝐺) = min{π‘“π‘π‘Ÿ(𝑆)}, where the minimum is established over all π‘π‘Ÿ-sets 𝑆 in 𝐺. The cardinality of a minimum forcing subset of 𝑆 is the forcing circular number” of 𝑆. Example 2.2. The only two π‘π‘Ÿ-sets of the graph G displayed in Figure 2.1 are 𝑆1 = {a𝑣1, a𝑣4, 𝑣5} and 𝑆2 = {a𝑣1, a𝑣4, 𝑣6} such that π‘“π‘π‘Ÿ(𝑆1) = π‘“π‘π‘Ÿ(𝑆2) = 1 and π‘“π‘π‘Ÿ(𝐺) = 1. Figure 2.1 𝑣5 𝑣1 𝑣3 𝑣2 𝑣6 𝑣4 𝐺 Figure 2.1 𝑣9 𝑣7 𝑣8 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 221 https://internationalpubls.com Observation 2.3. For each graph 𝐺 that is connected, 0 ≀ π‘“π‘π‘Ÿ(𝐺) ≀ π‘π‘Ÿ(𝐺). Remark 2.4. Observation 2.3 has sharp bounds. For 𝐺 = 𝑃3, π‘“π‘π‘Ÿ(𝐺) = 0. For 𝐺 = 𝐢4 with vertex set 𝑉(𝐺) = {𝑣1, 𝑣2, 𝑣3, 𝑣4}, 𝑆1 = {a𝑣1, a𝑣2}, 𝑆2 = {a𝑣2, a𝑣3}, 𝑆3 = {𝑣3, 𝑣4}, 𝑆4 = {𝑣7, a𝑣4}, 𝑆5 = {𝑣1, a𝑣3} and 𝑆6 = {a𝑣2, a𝑣4} are the only six π‘π‘Ÿ-sets of a𝐺 there exists π‘“π‘π‘Ÿ(𝑆𝑖) = 2, 1 ≀ 𝑖 ≀ 6 so that π‘“π‘π‘Ÿ(𝐺) = π‘π‘Ÿ(𝐺) = 2. Additionally, the limitations in Observation 2.3 may be extremely rigorous. The graph 𝐺 shown in Figure 2.1 has two values: π‘π‘Ÿ(𝐺) = 2 and π‘“π‘π‘Ÿ(𝐺) = 1. Hence 0 < π‘“π‘π‘Ÿ(𝐺) < π‘π‘Ÿ(𝐺). Theorem 2.5. Consider a connected graph, 𝐺. Following that i) π‘“π‘π‘Ÿ(𝐺) = 0 iff 𝐺 has a unique minimum π‘π‘Ÿ-set of 𝐺. ii) π‘“π‘π‘Ÿ(𝐺) = 1 iff 𝐺 possesses a minimum of two π‘π‘Ÿ-sets, at least one of which is a distinct π‘π‘Ÿ-et that includes one of its elements. iii) π‘“π‘π‘Ÿ(𝐺) = π‘π‘Ÿ(𝐺) iff any proper subset of 𝐺 that is not contained in any π‘π‘Ÿ-set is the unique minimal π‘π‘Ÿ-set of G. Definition 2.6. A vertex 𝑣 β€œof a connected graph 𝐺. If 𝑣 belongs to each π‘π‘Ÿ-set of 𝐺, then 𝑣(𝐺) is considered to be a circular vertex of 𝐺. Example 2.7. For the graph 𝐺 shown in Figure 2.2, the set of all circular vertices of 𝐺 is represented by {𝑣1, 𝑣3, 𝑣5} since 𝑆1 = {e𝑣1, e𝑣3, 𝑣5, e𝑣6} and 𝑆2 = {e𝑣1, 𝑣3, 𝑣5, 𝑣9} are the only two π‘π‘Ÿ-sets” of e𝐺. Figure 2.2 Theorem 2.8. Let π‘Š be the set of all circular vertices of connected graph 𝐺. Then π‘“π‘π‘Ÿ(e𝐺) ≀ π‘π‘Ÿ(e𝐺)– |eπ‘Š|. Remark 2.9. The bounds in Theorem 2.8 are precise. Regarding the graph G shown in Figure 2.2, |eπ‘Š| = 3, π‘π‘Ÿ(𝐺) = 4 and π‘“π‘π‘Ÿ(𝐺) = 1. Thus π‘“π‘π‘Ÿ(𝐺) = π‘π‘Ÿ(𝐺)– |π‘Š|. Moreover, the bounds in Theorem 2.8 may be rigid. With respect to graph G displayed in Figure 2.3, 𝑆1 = {a𝑣1, a𝑣4, a𝑣5, a𝑣7}, 𝑆2 = {a𝑣1, a𝑣4, a𝑣5, a𝑣8}, 𝑆3 = {𝑣1, a𝑣4, a𝑣5, 𝑣9}, 𝑆4 = {a𝑣2, a𝑣4, a𝑣5, a𝑣7}, 𝑆5 = {a𝑣2, 𝑣4, a𝑣5, 𝑣8} and 𝑆6 = 𝑣6 𝑣1 𝑣3 𝑣2 𝑣7 𝑣4 𝑣9 𝑣10 𝑣8 𝑣5 𝑣 𝑣13 𝑣12 𝑣11 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 222 https://internationalpubls.com {𝑣2, 𝑣4, 𝑣5, 𝑣9} are the six π‘π‘Ÿ-sets of 𝐺 so that {𝑣1, 𝑣4, 𝑣5} is the set of all circular vertices of e𝐺 there exists π‘“π‘π‘Ÿ(𝐺) = 1 and π‘π‘Ÿ(𝐺) = 3. Figure 2.3 Theorem 2.10. β€œFor the complete bipartite graph 𝐺 = eπΎπ‘Ÿ,𝑠, (1 ≀ eπ‘Ÿ ≀ 𝑠), π‘“π‘π‘Ÿ(𝐺) = { 0 𝑖𝑓 π‘Ÿ = 1, e𝑠 β‰₯ 2 2 𝑖𝑓 2 ≀ eπ‘Ÿ ≀ e𝑠 Proof. Let π‘ˆ = {e𝑒1, e𝑒2, . . . , eπ‘’π‘Ÿ} and π‘Š = {e𝑀1, e𝑀2, . . . , e𝑀𝑠} be the bipartite sets of 𝐺. For 𝑠 β‰₯ 2 and π‘Ÿ = 1, 𝑆 = π‘Š is the distinct π‘π‘Ÿ-set of e𝐺 so that π‘“π‘π‘Ÿ(e𝐺) = 0. Hence 2 ≀ eπ‘Ÿ ≀ e𝑠. Let 𝑀 ∈ π‘Š and 𝑒 ∈ π‘ˆ. Such that 𝑆 = {𝑒,𝑀} is a π‘π‘Ÿ-set of 𝐺.” Since this is true for all eπ‘’βˆˆ eπ‘ˆ and eπ‘€βˆˆ eπ‘Š, 𝑆 is not unique π‘π‘Ÿ-set of 𝐺 containing 𝑒 or 𝑀. Therefore π‘“π‘π‘Ÿ(𝐺) = 2. As this holds β€œtrue for every π‘π‘Ÿ- sets 𝑆 of e𝐺, π‘“π‘π‘Ÿ(e𝐺) = 2. Theorem 2.11. For the non-trivial tree 𝑇, π‘“π‘π‘Ÿ(𝑇) = 0. Proof. Considering 𝑆 to be the collection of all end vertices in 𝐺, 𝑆 is the only π‘π‘Ÿ-set in 𝐺 such that π‘“π‘π‘Ÿ(𝐺) = 0. Theorem 2.12. β€œFor the cycle e𝐺 = e𝐢𝑛,(e𝑛β‰₯4), π‘“π‘π‘Ÿ(e𝐺) = 2.” Proof. Let π‘₯ and 𝑦 represent any two vertices of 𝐺. There exists 𝑆 = {π‘₯, 𝑦} is a π‘π‘Ÿ-set of 𝐺. Hence eπ‘₯ and e𝑦 are arbitrary, 𝑆 is not a unique π‘π‘Ÿ-set containing π‘₯ or 𝑦. Therefore π‘“π‘π‘Ÿ(𝐺) = 2. As this holds true for all π‘π‘Ÿ-sets 𝑆 of 𝐺 ” therefore π‘“π‘π‘Ÿ(𝐺) = 2. Theorem 2.13. β€œFor the wheel e𝐺 = e𝐾 1 +𝐢 π‘›βˆ’1 ,(e𝑛β‰₯5), π‘“π‘π‘Ÿ(e𝐺) = 1.” Proof. Assume that π‘₯ represents the central vertex of 𝐺 and eπΆπ‘›βˆ’1 be 𝑣1, 𝑣2, … , π‘£π‘›βˆ’1, 𝑣1. Then 𝑆𝑖 = {π‘₯, 𝑣𝑖} (1 ≀ 𝑖 ≀ 𝑛 βˆ’ 1) and 𝑆 = {𝑒, 𝑣} where 𝑒 and 𝑣 are any two vertices in πΆπ‘›βˆ’1 are the π‘π‘Ÿ-sets of 𝐺. Now π‘“π‘π‘Ÿ(𝑆𝑖) = 1 (1 ≀ 𝑖 ≀ 𝑛 βˆ’ 1). Since 𝑒 and 𝑣 are arbitrary, 𝑆 is not a dsitinct π‘π‘Ÿ-set containing 𝑒 or 𝑣. Therefore π‘“π‘π‘Ÿ(𝐺) = 2. Hence it follows that π‘“π‘π‘Ÿ(𝐺) = 1. Theorem 2.14. β€œFor the fan graph 𝐹𝑛 = e𝐾1 + π‘ƒπ‘›βˆ’1, (e𝑛 β‰₯ 5), π‘“π‘π‘Ÿ(e𝐺) = 1.” Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 223 https://internationalpubls.com Proof. Suppose that π‘₯ represents the central vertex of of 𝐺 and 𝑉(π‘ƒπ‘›βˆ’1) = {𝑣1, 𝑣2, … , π‘£π‘›βˆ’1}. Then 𝑆𝑖 = {π‘₯, 𝑣𝑖} (1 ≀ 𝑖 ≀ 𝑛 βˆ’ 1) and 𝑆 = {𝑒, 𝑣} where 𝑒 and 𝑣 are any two vertices in π‘ƒπ‘›βˆ’1 are the π‘π‘Ÿ- sets of 𝐺. Now π‘“π‘π‘Ÿ(𝑆𝑖) = 1 (1 ≀ 𝑖 ≀ 𝑛 βˆ’ 1). Since 𝑒 and 𝑣 are arbitrary, 𝑆 is not a unique π‘π‘Ÿ-set containing 𝑒 or 𝑣. Therefore π‘“π‘π‘Ÿ(𝐺) = 2. Hence it follows that π‘“π‘π‘Ÿ(𝐺) = 1. 3. The Forcing Geodetic Numbers and the Forcing Circular Number of a Graph The forcing geodetic numbers and the forcing circular number of a graph have no relationship, as the example below demonstrates. Example 3.1. The unique 𝑔-set of the graph 𝐺 shown in Figure 3.1 is indicated as , 𝑆 = {𝑣1, 𝑣4, 𝑣5}. Therefore 𝑓𝑔(𝐺) = 0. Also β€œπ‘†1 = {𝑣1, 𝑣4} and 𝑆2 = {𝑣1, 𝑣5} are the only two π‘π‘Ÿ-sets of 𝐺 such that π‘“π‘π‘Ÿ(𝐺) = 1. Thus 𝑓𝑔(𝐺) < π‘“π‘π‘Ÿ(𝐺). Example 3.2. The unique π‘π‘Ÿ-set of the graph 𝐺 shown in Figure 3.2 is represented as 𝑆 = {𝑣1, 𝑣2}. Therefore π‘“π‘π‘Ÿ(𝐺) = 0. Also 𝑆1 = {𝑣1, 𝑣3, 𝑣6} and 𝑆2 = {𝑣1, 𝑣4, 𝑣6} are the only 𝑔-sets of 𝐺 so that 𝑓𝑔(𝐺) = 1.” Thus 𝑓𝑔(𝐺) > π‘“π‘π‘Ÿ(𝐺). Theorem 3.3. In a connected graph 𝐺, 𝑓𝑔(𝐺) = π‘Ž and π‘“π‘π‘Ÿ(𝐺) = 0 exist for each integer π‘Ž β‰₯ 0. Proof. Assume that 𝑃: 𝑒, 𝑣, 𝑀, π‘₯ is an order four path. Consider 𝑃𝑖: 𝑒𝑖 , 𝑣𝑖 (1 ≀ 𝑖 ≀ π‘Ž) represent an identical pair of vertices. Let 𝐺 be the graph generated by adding the edges 𝑣𝑒𝑖 and 𝑀𝑣𝑖 to 𝑃 and 𝑃𝑖 (1 ≀ 𝑖 ≀ π‘Ž). The figure 3.3 displays the graph 𝐺. 𝐺 Figure 3.1 𝑣1 𝑣2 𝑣3 𝑣4 𝑣5 𝐺 Figure 3.2 𝑣1 𝑣2 𝑣3 𝑣4 𝑣5 𝑣7 𝑣6 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 224 https://internationalpubls.com We first establish that 𝑓𝑔(𝐺) = π‘Ž. Let 𝑍 = {𝑒, π‘₯} represent all of 𝐺's end vertices. 𝑍 is a subset of every 𝑔-set in 𝐺, according to Theorem 1.1. For (1 ≀ 𝑖 ≀ π‘Ž), consider 𝐻𝑖 = {𝑒𝑖 , 𝑣𝑖}. It is easily shown that 𝑔(𝐺) β‰₯ π‘Ž since every vertex in the 𝑔-set of 𝐺 contains exactly one vertex from each 𝐻𝑖(1 ≀ 𝑖 ≀ π‘Ž). Let 𝑆 = 𝑍 βˆͺ {𝑒1, 𝑒2, … , π‘’π‘Ž}. As a result, 𝑆 is a 𝑔-set of 𝐺 and 𝑔(𝐺) = π‘Ž + 2, as 𝐼[𝑆] = 𝑉(𝐺). For every 𝑔-set of 𝐺 contains a subset, 𝑍. By Theorem 1.2, 𝑓𝑔(𝐺) ≀ 𝑔(𝐺) βˆ’ |𝑍| = π‘Ž + 2 βˆ’ 2 = π‘Ž. Therefore 𝑓𝑔(𝐺) ≀ π‘Ž. Considering that 𝑔(a𝐺) = aπ‘Ž + 2 and that 𝑍 exists in every 𝑔-set of 𝐺, following that each 𝑔-set of 𝐺, if so, has the form 𝑆 = 𝑍 βˆͺ {a𝑐1, a𝑐2, … , aπ‘π‘Ž}, where 𝑐𝑖 ∈ 𝐻𝑖(1 ≀ 𝑖 ≀ π‘Ž). Given |𝑇| < π‘Ž, let 𝑇 be any proper subset of 𝑆. After that, a𝑐𝑗 (1 ≀ a𝑗 ≀ aπ‘Ž) is a vertex such that a𝑐 𝑗 βˆ‰ a𝑇. Assume that 𝑏𝑗, a vertex of 𝐻𝑗, is distinct from a𝑐 𝑗 . Consequently, a𝑆 1 = ( aπ‘†βˆ’ { a𝑐 𝑗} ) βˆͺ { a𝑏 𝑗 } is a g-set that properly contains 𝑇. As a result, 𝑇 is not a forcing subset of S. For every minimum 𝑔-set of 𝐺, this holds true. Therefore 𝑓𝑔(𝐺) = π‘Ž. Next we prove that π‘“π‘π‘Ÿ(𝐺) = 0. Since 𝑍 is the distinct π‘π‘Ÿ-set of 𝐺, π‘“π‘π‘Ÿ(𝐺) = 0. Figure 3.3 Theorem 3.4. For every integer π‘Ž β‰₯ 0, β€œthere exists a connected graph 𝐺 such that 𝑓𝑔(𝐺) = 0 and π‘“π‘π‘Ÿ(𝐺) = π‘Ž. For every integer π‘Ž β‰₯ 0, there exists a connected graph 𝐺 such that 𝑓𝑔(𝐺) = π‘Ž and π‘“π‘π‘Ÿ(𝐺) = π‘Ž. Proof. Let 𝑃′: 𝑀1, 𝑀2, 𝑀3 be a path of order 3, and β€œconsider 𝑃𝑖: 𝑑1, 𝑑2, 𝑑3, 𝑑4, 𝑑5 be a path of order 5. Let 𝑃𝑖e: π‘Ÿπ‘–e, 𝑠𝑖e (1 ≀ 𝑖 ≀ π‘Ž) be an order 2 replica of the path. Let 𝐺e be the graph created by adding the edges 𝑑2𝑀1, 𝑑2𝑀2, 𝑑4𝑀2, 𝑑4𝑀3, 𝑑2π‘Ÿπ‘– (1 ≀ 𝑖 ≀ π‘Ž) and 𝑑4𝑆𝑖 (1 ≀ 𝑖 ≀ π‘Ž) to 𝑃′ and 𝑃𝑖 (1 ≀ 𝑖 ≀ π‘Ž). The Figure 3.4 displays the graph 𝐺. First, we establish that π‘“π‘π‘Ÿ(𝐺) = π‘Ž. Let the set of all of 𝐺's end vertices be 𝑍 = {𝑑1, 𝑑5}. Such that 𝑍 is therefore a subset of each π‘π‘Ÿ-set of 𝐺 according to Theorem 1.1. 𝐻𝑖e: {π‘Ÿπ‘– e, 𝑠𝑖e} (1 ≀ 𝑖 ≀ π‘Ž) be given. Then, it is evident that π‘π‘Ÿ(𝐺) β‰₯ π‘Ž + 2 since every circular set of 𝐺 has at least one vertex from Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 225 https://internationalpubls.com each 𝐻𝑖 (1 ≀ 𝑖 ≀ π‘Ž). Let 𝑆 = 𝑍 βˆͺ {π‘Ÿ1, π‘Ÿ2, … , π‘Ÿπ‘Ž}. 𝐼𝐷𝑐[𝑆] = 𝑉(𝐺) in this case, indicating that 𝑆 is a circular set of 𝐺 and hence π‘π‘Ÿ(𝐺) = π‘Ž + 2. As 𝑍 is a subset of each π‘π‘Ÿ-set of 𝐺e, π‘“π‘π‘Ÿ(𝐺) ≀ π‘π‘Ÿ(𝐺) βˆ’ |𝑍| = π‘Ž + 2 βˆ’ 2 = π‘Ž, according to Theorem 2.3. Consequently, π‘“π‘π‘Ÿ(𝐺) ≀ π‘Ž. Given that π‘π‘Ÿ(𝐺) = π‘Ž = 2, Furthermore, it is evident that every π‘π‘Ÿ-set of 𝐺 that contains 𝑍 has the form 𝑆 = 𝑍e βˆͺ {𝑐1 e, 𝑐2, … , π‘π‘Ž}, where 𝑐𝑖 ∈ 𝐻𝑖 (1 ≀ 𝑖e ≀ π‘Ž). Given |𝑇| < π‘Že, let 𝑇 be any suitable subset of 𝑆. After that, 𝑐𝑗 (1 ≀ 𝑗e ≀ π‘Ž) is a vertex such that 𝑐𝑗 βˆ‰ 𝑇. Assume that 𝑏𝑗, a vertex of 𝐻𝑗, is different from 𝑐𝑗. Subseequently, 𝑆1 = (𝑆 βˆ’ {𝑐𝑗}) βˆͺ {𝑏𝑗} is a π‘π‘Ÿ-set that correctly contains 𝑇. 𝑇 is not a forced subset of 𝑆” as a result. For every minimum π‘π‘Ÿ-set of 𝐺, this is true. Consequently, π‘“π‘π‘Ÿ(𝐺) = π‘Ž. Next, we prove that 𝑓𝑔(𝐺) = π‘Ž. The representation of every extreme vertex in 𝐺 is 𝑍1 = 𝑍 βˆͺ {𝑀1, 𝑀3}. Theorem 1.1 states that every 𝑔-set in 𝐺 is a subset of 𝑍1. Give 𝐻𝑖: {π‘Ÿπ‘– e, 𝑠𝑖 e} (1 ≀ 𝑖e ≀ π‘Ž). Since every 𝑔-set of 𝐺 contains at least one vertex from every 𝐻𝑖(1 ≀ 𝑖e ≀ π‘Ž), it is easy to demonstrate that 𝑔(𝐺) β‰₯ π‘Ž + 4. 𝑆 = 𝑍1 βˆͺ {π‘Ÿ1, π‘Ÿ2, … , π‘Ÿπ‘Ž} is assumed. Consequently, since 𝐼[𝑆] = 𝑉(𝐺), 𝑆 is a 𝑔-set of 𝐺 and 𝑔(𝐺) = π‘Ž + 4. All of the 𝑔-sets in 𝐺 have a subset called 𝑍1. The 𝑓𝑔(𝐺) ≀ π‘Ž Theorem applies. 𝑍1 appears in every 𝑔-set of 𝐺, and since 𝑔(𝐺) = π‘Ž + 4, it follows that every 𝑔-set of 𝐺 has the form 𝑆 = 𝑍1 βˆͺ {𝑐1, 𝑐2 e, … , π‘π‘Ž e}, where 𝑐𝑖 ∈ 𝐻𝑖(1 ≀ 𝑖e ≀ π‘Že). Let 𝑇 be any appropriate subset of 𝑆 such that |𝑇| < π‘Ž. After that, a vertex such that is 𝑐𝑗 βˆ‰ 𝑇 is 𝑐𝑗 (1 ≀ 𝑗 ≀ π‘Ž). Presume that 𝑑𝑗, one of 𝐻𝑗 's vertices, is distinct from 𝑐𝑗. Consequently, a 𝑔-set that suitably contains 𝑇 is 𝑆1 = (𝑆 βˆ’ {𝑐𝑗}) βˆͺ {𝑏𝑗}. As such, 𝑇 is not a forced subset of 𝑆. This is valid for any smallest 𝑔-set of 𝐺. As a result, 𝑓𝑔(𝐺) = π‘Ž. Figure 3.4 Theorem 3.5. Let G be a connected graph. For every integer π‘Ž β‰₯ 0, and 𝑏 β‰₯ 0, there exists 𝑓𝑔(𝐺) = π‘Ž and π‘“π‘π‘Ÿ(𝐺) = 𝑏. Proof. Case (i) π‘Ž = 0, 𝑏 β‰₯ 1. The graph produced in Theorem 3.3 meets the requisite requirement. Case (ii) π‘Ž β‰₯ 1, 𝑏 = 0. The graph constructed Theorem 3.3, satisfies the required condition. Case (iii) π‘Ž = 𝑏 β‰₯ 1. The graph constructed Theorem 3.4, satisfies the required condition. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 226 https://internationalpubls.com Case (iv) 0 < π‘Ž < 𝑏. Consider the graph 𝐺 given in Figure 3.5. We first establish that 𝑓𝑔(𝐺) = π‘Ž. The set of all extreme vertex of 𝐺 is denoted by 𝑍 = {𝑑, 𝑀1, 𝑀3, π‘₯1, π‘₯2, … , π‘₯π‘βˆ’π‘Ž, 𝑦1, 𝑦2, … , π‘¦π‘βˆ’π‘Ž}. Thus 𝑍 is a subset of each geodetic set in 𝐺, according to Theorem 3.4. 𝐻𝑖: {π‘Ÿπ‘–, 𝑠𝑖} (1 ≀ 𝑖 ≀ π‘Ž) be given. Every geodetic set of β€œπΊ has exactly one vertex from each 𝐻𝑖 (1 ≀ 𝑖e ≀ π‘Ž), as can be seen easily. As a result, (𝐺) β‰₯ 3 + 𝑏 βˆ’ π‘Ž + 𝑏 βˆ’ π‘Ž + π‘Ž = 2𝑏 βˆ’ π‘Ž + 3. Let 𝑆e = 𝑍 βˆͺ {eπ‘Ÿ1, eπ‘Ÿ2, … , eπ‘Ÿπ‘Ž}. Thus, 𝑆 is a geodetic set of 𝐺e since 𝐼[e𝑆] = 𝑉(𝐺). Consequently, 𝑔(e𝐺) = 2𝑏 βˆ’ π‘Ž + 3. 𝑓𝑔(𝐺) ≀ 𝑔(𝐺) βˆ’ |𝑍| = 2𝑏 βˆ’ π‘Ž + 3 βˆ’ (2𝑏 βˆ’ 2π‘Ž + 3) = π‘Ž, according to the theorem. Given that 𝑔(𝐺) = 2𝑏 βˆ’ π‘Ž + 3 and that 𝑍1 appears in every 𝑔-set of 𝐺, it is clear that every 𝑔-set of 𝐺 has the form 𝑆 = 𝑍 βˆͺ {e𝑐1, e𝑐2, … , eπ‘π‘Ž}where 𝑐𝑖 ∈ 𝐻𝑖 (1 ≀ 𝑖 ≀ eπ‘Ž). Given |𝑇| < π‘Ž, let e𝑇 be any suitable subset of 𝑆. After that, 𝑐𝑗 (1 ≀ 𝑗 ≀ π‘Ž)” is a vertex such that e𝑐𝑗 βˆ‰ 𝑇. Assume that e𝑏 𝑗 , a vertex of 𝐻𝑗, is different from 𝑐𝑗. Subsequently, 𝑆1 = (𝑆 βˆ’ {𝑐𝑗}) βˆͺ {𝑏𝑗} is a 𝑔-set that correctly contains e𝑇. Such that e𝑇 is not a forced subset of 𝑆 as a result. For every smallest 𝑔-set of 𝐺, this holds true. Hence, 𝑓𝑔(𝐺) = π‘Ž. Figure 3.5 We then demonstrate that π‘“π‘π‘Ÿ(𝐺) = 𝑏. Let 𝑍1 = {𝑑} represent 𝐺's end vertex. Such that 𝑍 is a subset of every π‘π‘Ÿ-set in 𝐺 according to Theorem 1.1. 𝑄𝑗: {π‘₯𝑗 , 𝑦𝑗} (1 ≀ 𝑗 ≀ 𝑏 βˆ’ π‘Ž) be given. Every circular set of 𝐺 has exactly one vertex from every 𝑄𝑗 (1 ≀ 𝑗 ≀ 𝑏 βˆ’ π‘Ž) and exactly one vertex from every 𝐻𝑖 (1 ≀ 𝑖 ≀ π‘Ž), as can be seen easily. Therefore, π‘π‘Ÿ(𝐺) β‰₯ 1 + π‘Ž + 𝑏 βˆ’ π‘Ž = 𝑏 + 1. Assume that 𝑆 = 𝑍 βˆͺ {π‘Ÿ1, π‘Ÿ2, … , π‘Ÿπ‘Ž} βˆͺ {π‘₯1, π‘₯2, … , π‘₯π‘βˆ’π‘Ž}. As a result, 𝑆 is a circular set of 𝐺 since 𝐼[𝑆] = 𝑉(𝐺). Consequently, π‘π‘Ÿ(𝐺) = 𝑏 + 1. Therefore π‘“π‘π‘Ÿ(𝐺) ≀ π‘π‘Ÿ(𝐺) βˆ’ |𝑍| = 𝑏 + 1 βˆ’ 1 = 𝑏, according to the theorem. Every circular set of 𝐺, of which 𝑍 is a subset, has the form π‘Š1 = 𝑍 βˆͺ {𝑐1, 𝑐2, … , π‘π‘Ž} βˆͺ {𝑑1, 𝑑2, … , π‘‘π‘βˆ’π‘Ž}, where 𝑑𝑗 ∈ 𝑄𝑗 (1 ≀ 𝑗 ≀ 𝑏 βˆ’ π‘Ž) and 𝑐𝑖 ∈ 𝐻𝑖 (1 ≀ 𝑖 ≀ π‘Ž). Given |𝑇| < 𝑏, let 𝑇 be any proper subset of π‘Š1. After that, 𝑐𝑗, 𝑑𝑗 βˆ‰ 𝑇 since there are vertices 𝑐𝑗 ∈ 𝐻𝑖 and 𝑑𝑗 ∈ 𝑄𝑗. Assume that 𝑓𝑗 is a vertex of 𝑄𝑗 that is separate from 𝑑𝑗 and that 𝑒𝑖 is a vertex of 𝐻𝑖 apart from 𝑐𝑖. π‘Š2 = Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 227 https://internationalpubls.com (π‘Š1 βˆ’ {𝑐𝑗, 𝑑𝑗}) βˆͺ {𝑒𝑖, 𝑓𝑗} is a π‘π‘Ÿ-set that correctly contains 𝑇 in this case. Such that 𝑇 is not a forced subset of π‘Š2 as a result. For every minimum π‘π‘Ÿ-set of 𝐺, this is true. Thus, π‘“π‘π‘Ÿ(𝐺) = 𝑏. Case (v) 0 < 𝑏 < π‘Ž. First, we establish that π‘“π‘π‘Ÿ(𝐺) = 𝑏. Assume 𝑍 = {𝑑1, w}. Thus 𝑍 is therefore a subset of 𝐺's π‘π‘Ÿ-set. Assume β€œπ»π‘–e: {π‘Ÿπ‘– e, e𝑠𝑖} (1 ≀ 𝑖e ≀ 𝑏)be given. It is simple to see that π‘π‘Ÿ(𝐺) β‰₯ 𝑏 + 2 as every circular set of 𝐺 has at least one vertex from each 𝐻𝑖(1 ≀ 𝑖 ≀ π‘Ž). Consider 𝑆 = 𝑍1 βˆͺ {eπ‘Ÿ1, eπ‘Ÿ2, … , eπ‘Ÿπ‘}. Consequently, 𝑆 is a circular set of e𝐺 since 𝐼[e𝑆] = 𝑉(𝐺), and π‘π‘Ÿ(𝐺) = 𝑏 + 2. Given that e𝑍 is a subset of each 𝐺 π‘π‘Ÿ-set. According to Theorem π‘“π‘π‘Ÿ(𝐺) ≀ π‘π‘Ÿ(𝐺) βˆ’ |𝑍| = +2 βˆ’ 2 = e𝑏. Thus, π‘“π‘π‘Ÿ(𝐺) ≀ 𝑏. It is clear that every π‘π‘Ÿ-set of 𝐺 is of the type 𝑆 = e𝑍1 βˆͺ {𝑐1, e𝑐2, … , eπ‘π‘Ž}, where 𝑐𝑖 ∈ 𝐻𝑖(1 ≀ 𝑖 ≀ π‘Ž), since π‘π‘Ÿ(𝐺) = 𝑏 + 2 and every π‘π‘Ÿ-set of 𝐺 contains 𝑍. Given |𝑇| < π‘Ž, let 𝑇 be any proper subset of 𝑆. After that, e𝑐𝑗 (1 ≀ e𝑗 ≀ eπ‘Ž) is a vertex such that e𝑐 𝑗 βˆ‰ e𝑇. Assume that 𝑏𝑗, a vertex of 𝐻𝑗,” is different from 𝑐𝑗. Following that, 𝑆1 = (𝑆 βˆ’ {𝑐𝑗}) βˆͺ {𝑏𝑗} is a π‘π‘Ÿ-set that correctly contains e𝑇. Such that 𝑇 is not a forced subset of 𝑆 as a result. For every minimum π‘π‘Ÿ-set of 𝐺, this is true. Hence, π‘“π‘π‘Ÿ(𝐺) = π‘Ž. We then demonstrate that 𝑓𝑔(𝐺) = π‘Ž. Let 𝑍 = {𝑑1, 𝑑3, 𝑀1, 𝑀3} represent all of 𝐺's extreme vertices. Consider 𝑍 is a subset of each geodetic set in 𝐺, according to Theorem 3.4. 𝑍 should be 𝑍 = 𝑍1 βˆͺ {𝑀}. Therefore 𝑍1 is clearly a subset of each and every geodetic set in 𝐺. Assume 𝑄𝑗: {𝑒𝑗 , 𝑣𝑗} (1 ≀ 𝑗 ≀ π‘Ž βˆ’ 𝑏). Every geodetic set of 𝐺 has at least one vertex from each of the 𝐻𝑗 (1 ≀ 𝑗 ≀ π‘Ž) and each of the 𝑄𝑗, as can be clearly recognised; so, 𝑔(𝐺) β‰₯ 4 + π‘Ž βˆ’ 𝑏 + 𝑏 = 4 + π‘Ž. Assume π‘Š = 𝑍1 βˆͺ {π‘Ÿ1, π‘Ÿ2, … , π‘Ÿπ‘ , 𝑒1, 𝑒2, … , π‘’π‘Žβˆ’π‘}. Consequently, 𝑔(𝐺) = π‘Ž + 4 since 𝐼[π‘Š] = 𝑉(𝐺) and π‘Š is a geodetic set of 𝐺. 𝑓𝑔(𝐺) ≀ 𝑔(𝐺) βˆ’ |𝑍| = π‘Ž + 4 βˆ’ 4 = π‘Ž according to Theorem 1.2. It is evident that every 𝑔- set of 𝐺 is of the form π‘Š1 = 𝑍 βˆͺ {e𝑐1, e𝑐2, … , e𝑐𝑏} βˆͺ {𝑑1, 𝑑2, … , π‘‘π‘Žβˆ’π‘} since 𝑍 is a subset of every 𝑔-set of e𝐺. In this case, 𝑑𝑗 ∈ 𝑄𝑗 (1 ≀ 𝑗 ≀ π‘Ž βˆ’ 𝑏) and 𝑐𝑖 ∈ 𝐻𝑖 (1 ≀ 𝑖 ≀ 𝑏). Given |𝑇| < 𝑏, let 𝑇 be any proper subset of π‘Š1. After that, 𝑐𝑗 , 𝑑𝑗 βˆ‰ 𝑇 since there are vertices 𝑐𝑖 ∈ 𝐻𝑖 and 𝑑𝑗 ∈ 𝑄𝑗. Assume 𝑄𝑗 is a vertex of 𝐻𝑖 that is different from 𝑑𝑗 and 𝑐𝑖. π‘Š2 = (π‘Š1 βˆ’ {𝑐𝑗, 𝑑𝑗}) βˆͺ {𝑒𝑗, 𝑓𝑗} is a π‘π‘Ÿ-set that correctly contains 𝑇 in this case. Such that 𝑇 is not a forced subset of 𝑆 as a result. For every minimum π‘π‘Ÿ-set of 𝐺, this is true. Therefore, 𝑓𝑔(𝐺) = π‘Ž. Figure 3.6 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 228 https://internationalpubls.com References [1] F. Buckley and F. Harary, Distance in Graphs, Addision-Weseely, Reading MA, (1990). [2] G. Chartrand and P. Zhang, The forcing geodetic number of a graph, Discuss. Graph Theory, 19, (1999), 45-58. [3] G. Chartrand, F. Harary and P. Zhang, On the geodetic number of a graph, Networks, 39(1), (2002), 1 - 6. [4] G. Chartrand, E. M. Palmer and P. Zhang, The geodetic number of a graph, A Survey, Congressus Numerantium, 156, (2002), 37 - 58. [5] G. Chartrand, L. Johns and P. Zang, Detour Number of graph, Utilitas Mathematics, 64 (2003), 97-113. [6] G. Chartrand, H. Escuadro and P. Zhang, Distance in Graphs, Taking the Long View, AKCE J. Graphs and Combin., 1(1) (2004), 1-13. [7] G. Chartrand, H. EScuadro and B. 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