Independent Fixed Connected Geodetic Number of a Graph

: In this paper we introduce the concept of independent fixed connected geodetic number and investigate its behaviours on some standard graphs. Lower and upper bounds are found for the above number and we characterize the suitable graphs achieving these bounds. We also define two new parameters connected geo-independent number and upper connected geo-independent number of a graph. Few characterization and realization results are formulated for the new parameters. Finally an open problem is posed.


Introduction
The introduction of Graph Theory is a revolution in the field of Mathematics.Various concepts were made easily understandable by its simple expression through graphical models.By a graph G we mean V, the set of vertices; E, the set of edges together with a binary operation of association.We refer to [1][2][3] for basic graph theoretic terms.In G, a shortest x − y path is also known as x − y geodesic.The distance d(x, y) is defined as the number of edges of an x − y geodesic in G.For any two vertices x and y in G, the closed interval I[x, y] is the collection of vertices on an x − y geodesic.The closed interval I[S , S ′ ], where S , S ′ ⊆ V(G), is defined as the union of subintervals I[x, y] for some x ∈ S and y ∈ S ′ .i.e., I[S , S ′ ] = x∈S ,y∈S ′ I[x, y].A vertex v in G is called an extreme vertex or simplicial vertex if the subgraph induced by its adjacent vertices is complete.
A set S ⊆ V(G) is called a geodetic set or geodomination set if every vertex of G is on some x − y geodesic where x, y ∈ S .The minimum cardinality of a geodetic set of G is called as the geodetic number of G, denoted by g(G) [4][5][6][7][8].If S is a geodetic set of G and ⟨S ⟩ is connected, then S is called the connected geodetic set of G. Its minimum order is named as the connected geodetic number of G, denoted by cg(G).A connected geodetic set of cardinality cg(G) is called a cg-set of G [9].Again parameters upper connected geodetic number and forcing connected geodetic number were defined and investigated in [10].Santhakumaran and Titus first introduced the vertex geodomination number in [11] and further studied in [12,13].For any vertex x in G, a set S ⊆ V(G) is called an x-geodominating set of G if every vertex v in G is on an x − y geodesic for some y in S .The minimum cardinality of an x-geodominating set of G is defined as the x-geodomination number of G, denoted by g x (G).An x-geodominating set of cardinality g x (G) is called a g x -set of G.A connected x-geodominating set of G is an x-geodominating set S such that ⟨S ⟩ is connected.The minimum cardinality of a connected x-geodominating set of G is the connected x-geodomination number of G and is denoted by cg x (G).A connected x-geodominating set of cardinality cg x (G) is called a cg x -set of G [14].
Let e = xy be any edge of a connected graph G of order atleast 3. A set S of vertices of G is an e-geodominating set of G if every vertex of G is either on an x − u geodesic or on an y − u geodesic for some element u in S .The minimum cardinality of an e-geodominating set of G is defined as the egeodomination number of G and is denoted by g e (G) or g xy (G).An e-geodominating set of cardinality g e (G) is called a g e -set of G [15].
It is clear that x-geodominating set of G is obtained by fixing a vertex x in G and e-geodominating set of G is obtained by fixing an edge e = xy in G. Based on these concepts we defined a new parameter called independent fixed geodomination number (or independent fixed geodetic number) in [16].Let S be an independent set of a connected graph G of order atleast 2. Let S ′ be a subset of V(G).If each vertex v in G is on an x − y geodesic for some x ∈ S and y ∈ S ′ , then S ′ is an S -fixed geodetic set of G .The S -fixed geodetic number g s (G) of G is the minimum cardinality of an S -fixed geodetic set of G.The independent fixed geodetic number of G is g i f (G) = min {g s (G)}, where the minimum is taken over all independent sets S in G.An independent fixed geodetic set of cardinality g i f (G) is described as g i f -set of G.We too further proceed to infer about connectedness of an S -fixed geodetic set of G, where S is an independent set in a connected graph G.In the computation of independent fixed connected geodetic number, the succeeding theorems will be employed.
Theorem 2. [16] Let G be a connected graph.Then g i f (G) = 1 if and only if there is an independent set S and its eccentric vertex y such that every vertex of G is on an x − y geodesic for some x ∈ S .Theorem 3. [16] For any complete graph K p , g i f (K p ) = p − 1.

Independent fixed connected geodetic number
Definition 1.Let S be an independent set of a connected graph G of order atleast 2.An S -fixed connected geodetic set of G is an S -fixed geodetic set S ′ such that ⟨S ′ ⟩ is connected.The S -fixed connected geodetic number cg s (G) of G is the minimum cardinality of an S -fixed connected geodetic set of G.The independent fixed connected geodetic number cg i f (G) of G is defined as cg i f (G) = min {cg s (G)}, where the minimum is taken over all independent sets S in G.An independent fixed connected geodetic set of cardinality cg i f (G) is called a cg i f -set of G.
Example 1.Consider a graph G as shown in Figure 1.The Table 1 gives the independent sets S , their corresponding minimum S -fixed geodetic sets and minimum S -fixed connected geodetic sets, the S -fixed geodetic numbers g s (G) and the S -fixed connected geodetic numbers cg s (G).Then the independent fixed geodetic number of G is g i f (G) = min {g s (G)} = 2 and the independent fixed connected geodetic number of G is cg i f (G) = min {cg s (G)} = 3.
Proof.For any independent set S in a connected graph G, every S -fixed connected geodetic set is an S -fixed geodetic set of G, we have g i f (G) ≤ cg i f (G).Then by Theorem 1, we have 1 We intend to characterize the graphs that realize the bounds in Theorem 4. For that we use the following definition.Definition 2. [16] Let G be a connected graph of order atleast 2. Let S ⊂ V(G) and y ∈ V(G) − S .The distance between the set S and the vertex y is d(S , y) = min{d(x, y) : x ∈ S }.The eccentricity of the set S is e(S ) = max{d(S , y) : y ∈ V(G) − S }.An eccentric vertex of S is a vertex v of G with d(S , v) = e(S ).
Theorem 5. Let G be a connected graph of order atleast 2. Then cg i f (G) = 1 if and only if there is an independent set S and its eccentric vertex y such that every vertex of G is on an x − y geodesic for some x ∈ S .
Proof.The result follows from Theorems 2 and 4. □ The following theorem gives cg i f (G) for some standard graphs.
(ii) If G = C p , then cg i f (G) is 1 or 2 according as p is even or odd.
(iii) If G = K 1 + ∪m j K j with G is neither a complete graph nor a star, then cg i f (G) = j − 1 or m j ( j − 1) + 1 according as j≥2 m j = 1 or j≥2 m j ≥ 2.
In view of Theorem 4, we proceed to characterize graphs G with cg i f (G) = p − 1.
Theorem 7. Let G be a connected graph of order p ≥ 2.
Hence cg i f (G) ≤ p − 2, but this leads to a contradiction.If both u 0 and u d are non-end vertices of G, then there exist atleast two u 0 − u d paths in G.If atleast one vertex other than u 0 and u d is common for any two u 0 −u d paths in G, then S = {u 0 , u d } is an independent set of G and S ′ = V(G)−S is an S -fixed connected geodetic set of G. Hence cg i f (G) ≤ p − 2, but this leads to a contradiction.Suppose that there exist two u 0 − u d paths, say P 1 and P 2 , has no common vertices other than u 0 and u d .Let z be an adjacent vertex of u 0 in but this leads to a contradiction.If z is not a cut vertex and {u 0 , z} is a vertex-cut of G, then there exists another u 0 − z path, say Q, of length atleast 2. Let w be an adjacent vertex For that we require the definition of a special graph K m ← P r → K n .Definition 3. The graph G = K m ← P r → K n is obtained from two complete graphs K m , K n and a path P r by joining every vertex in K m with an end vertex of P r and joining every vertex in K n with the other end vertex of P r .
The graph G = K m ← P r → K n is shown in Figure 2.  Now, let p ≥ 5. Since cg i f (G) = p − 2, by Theorem 7, we have G K p .First, we show that every vertex of G is either a cut vertex or a simplicial vertex.Incase there exists a vertex, say x, in G which is neither a cut vertex nor a simplicial vertex, then x lies on a cycle in G. Let C be a largest chordless cycle containing the vertex x in G.We consider three cases.Case (1) Length of the cycle C is more than 3 and degree of x is 2.
Let y and z be the adjacent vertices of x on C. Since C is a chordless cycle, y and z are nonsimplicial vertices of G. Subcase (1) deg y = deg z = 2. Let S = {x} be an independent set of G.If C is an even cycle, then y and z is on an x − x 1 geodesic, where x 1 is an eccentric vertex of x on C; and if C is an odd cycle, then y is on an x − x 1 geodesic and z is on an x − x 2 geodesic, where x not in both G 1 and G 2 .Let y 1 be a vertex farthest away from y in G 1 and let z 1 be a vertex farthest away from z in G 2 .Then it is vivid that S = {x, y 1 , z 1 } is an independent set of G and S ′ = V(G) − S is an S -fixed connected geodetic set of G. Hence cg i f (G) ≤ p − 3, but this leads to a contradiction.
If y and z are non-cut vertices of G and {y, z} is a vertex-cut of G − x, then y, x and z are the consecutive vertices of another chordless cycle, say C ′ , in G. Let z ′ x be an adjacent vertex of z on Then S = {x, z ′ } is an independent set of G and S ′ = V(G) − {x, y, z ′ } is an S -fixed connected geodetic set of G and so cg i f (G) ≤ p − 3, but this leads to a contradiction.
If z is not a cut vertex of G, then clearly S = {x} is an independent set of G and S ′ = V(G) − {x, y, z} is an S -fixed connected geodetic set of G. Hence cg i f (G) ≤ p − 3, but this leads to a contradiction.Case (2) Length of the cycle C is more than 3 and degree of x is more than 2. Subcase (1) {x, y} and {x, z} are non vertex-cuts of G. Then it is clear that S = {x} is an independent set of G and S ′ = V(G) − {x, y, z} is an S -fixed connected geodetic set of G. Hence cg i f (G) ≤ p − 3, but this leads to a contradiction.Subcase (2) {x, y} and {x, z} are vertex-cuts of G. Then G has two more cycles, say C 1 and C 2 , with xy an edge of C 1 and xz an edge of C 2 .Let x ′ y be an adjacent vertex of x on C 1 and let x ′′ z be an adjacent vertex of x on C 2 .If x ′ is a cut vertex of G, then take a = x ′ 1 , where x ′ 1 is a vertex farthest away from x ′ in a component H of G − x ′ with x not a vertex of H. Otherwise, take a = x ′ .Similarly, if x ′′ is a cut vertex of G, then take b = x ′′ 1 , where x ′′ 1 is a vertex farthest away from x ′′ in a component H 1 of G − x ′′ with x not a vertex of H 1 .Otherwise, take b = x ′′ .It is clear that S = {a, b} is an independent set of G and S ′ = V(G) − {x, a, b} is an S -fixed connected geodetic set of G.It implies cg i f (G) ≤ p − 3, but this leads to a contradiction.Subcase (3) {x, y} is a vertex-cut and {x, z} is not a vertex-cut of G ((or) {x, y} is not a vertex-cut and {x, z} is a vertex-cut of G).Then G has one more chordless cycle, say C 1 , with xy an edge of C 1 .Let x ′ y be an adjacent vertex of x on C 1 .If x ′ is adjacent to y in G and x ′ is not a cut vertex of G, then S = {x ′ } is an independent set of G and S ′ = V(G) − {x, x ′ , z} is an S -fixed connected geodetic set of G and so cg i f (G) ≤ p − 3, but this leads to a contradiction.If x ′ is adjacent to y in G and x ′ is a cut vertex of G, then by an argument similar to Subcase (3) of Case (1), S = {x, x ′′ } is an independent set of G and S ′ = V(G) − {x, x ′′ , z} is an S -fixed connected geodetic set of G. Hence cg i f (G) ≤ p − 3, but this leads to a contradiction.
Similarly, if x ′ is a non-adjacent vertex of y on C 1 and x ′ is a non-cut vertex of G, then S = {x} and S ′ = V(G) − {x, x ′ , z}.If x ′ is a non-adjacent vertex of y and x ′ is a cut vertex of G, then S = {x, x ′′ } and S ′ = V(G) − {x, x ′′ , z}, where x ′′ is a vertex farthest away from x ′ in a component H of G − x ′ with x not a vertex of H.In both cases, S is an independent set of G and S ′ is an S -fixed connected geodetic set of G and hence cg i f (G) ≤ p − 3, but this leads to a contradiction.Case (3) Length of the cycle C is 3. Then the graph H given in Figure 2.3 is a subgraph of G.If not, every block of G is either K 2 or K 3 and so every vertex of G is either a cut vertex or a simplicial vertex, which is a contradiction to our assumption.Here we take the vertex u as x and the cycle u, v, w, z, u in H as C and continue the procedure exactly similar to Case (1) and Case (2), we get cg i f (G) ≤ p − 3, but this leads to a contradiction.
Hence in all the three cases we get a contradiction and so every vertex in G is either a cut vertex or a simplicial vertex.Since cg i f (G) = p − 2, by Theorem 7, G is a non-complete graph.Hence G has atleast one cut vertex.Let Q = {u 1 , u 2 , . . ., u a } be the set of all cut vertices of G.We consider two cases.
Case (1) G has exactly one cut vertex, say u 1 .
Let G 1 , G 2 , . . ., G t (t ≥ 2) be the components of G − u 1 .If t ≥ 3, then S = {x 1 , x 2 , . . ., x t }, where Conversely, let G = P 3 or K m ← P r → K n (m, n ≥ 2 and r ≥ 1).If G = P 3 , then by Theorem 6(i), It is clear that any independent set S of G contains atmost one vertex from each K m and K n .Also, every S -fixed connected geodetic set of G contains atleast m − 1 vertices from K m and atleast n − 1 vertices from K n .Since m ≥ 2, n ≥ 2 and r ≥ 1, every vertex of P r is an element of any S -fixed connected geodetic set of G. Hence S ′ = V(G) − S is an S -fixed connected geodetic set of G and so cg i f (G) ≥ p − 2. Let S 1 = {x, y}, where x ∈ V(K m ) and y ∈ V(K n ), and let 1 is an S 1 -fixed connected geodetic set of G and so □ Based on Theorem 4, we have the following realization result.It is clear that any independent set of G contains atmost one vertex from the complete graph K a+1 .Also, atleast a vertices in K a+1 lie in every S -fixed geodetic set of G, where S is an independent set of G. Hence g i f (G) ≥ a.Let S = {v 1 , x} and S ′ = V(K a+1 ) − {x}, where x ∈ V(K a+1 ).Since a ≤ p − 3, S is an independent set of G and it is clear that every vertex of V(G) − S is on a v 1 − y geodesic for any y ∈ S ′ .Hence S ′ is an S -fixed geodesic set of G and so  It is clear that any independent set of G contains atmost one vertex from each complete graphs K 2 and K a .Also, atleast one vertex in K 2 and atleast a − 1 vertices in K a lie in every S -fixed geodetic set of G, where S is any independent set of G. Hence g i f (G) ≥ a.Let S = {v 1 , x, y} and let S ′ = (V(K 2 ) − {x}) ∪ (V(K a ) − {y}), where x ∈ V(K 2 ) and y ∈ V(K a ).Since a < b ≤ p − 3, S is an independent set of G and it is clear that every vertex of V(G) − S lies on a u − v geodesic for some u in S and v in S ′ .Hence S ′ is an S -fixed geodetic set of G and so g i f (G) = |S ′ | = a.But ⟨S ′ ⟩ is not connected and so cg i f (G) > a.Since every S -fixed geodetic set of G contains atleast one vertex from each complete graphs K 2 and K a , every S -fixed connected geodetic set of G contains the cut vertices We know that the diameter of any connected graph lies between its radius and two times of its radius.For that Ostrand [17] has given a realization result.Ostrand's theorem can be extended so that cg i f (G) can also be prescribed.The minimum (maximum) independent set required to form a cg i f -set of G is called a connected geo-independent set (upper connected geo-independent set) of G.The cardinality of a connected geo-independent set (upper connected geo-independent set) of G is called the connected geo-independent number (upper connected geo-independent number) of G and is denoted by cgin(G) (cgin + (G)).
The following result gives the connected geo-independent numbers and the upper connected geoindependent numbers of certain special classes of graphs.The following observation is an easy consequence of some of the previous results.
then by Theorem 7, cg i f (G) = 3 = p − 1, but this leads to a contradiction.If G = K 2 + K2 , then G has two simplicial vertices, say x and y.It is obvious that S = {x} is an independent set of G and S ′ = {y} is the minimum S -fixed connected geodetic set of G. Hence cg i f (G) = 1 = p − 3, but this leads to a contradiction.If G = K 1 + (K 1 ∪ K 2 ), K 1,3 , P 4 or C 4 , then by Theorem 6, cg i f (G) = 1 = p − 3, but this leads to a contradiction.
x 1 and x 2 are the eccentric vertices of x on C. Then it is clear that S ′ = V(G) − {x, y, z} is an S -fixed geodetic set of G. Since deg y = deg z = deg x = 2, ⟨S ′ ⟩ is connected.Hence S ′ is an S -fixed connected geodetic set of G and so cg i f (G) ≤ p − 3, but this leads to a contradiction.Subcase (2) deg y ≥ 3 and deg z ≥ 3. Since deg x = 2, G − x is a connected graph.If y and z are cut vertices of G, then let G 1 and G 2 be the components of G − y and G − z, respectively, with the vertex but this leads to a contradiction.If y and z are non-cut vertices of G and {y, z} is not a vertex-cut of G − x.It is clear that S = {x} is an independent set of G and S ′ = V(G) − {x, y, z} is an S -fixed connected geodetic set of G and so cg i f (G) ≤ p − 3, but this leads to a contradiction.If either y or z is a cut vertex of G, then by the arguments similar to the above, we get a contradiction.Subcase (3) deg y = 2 and deg z ≥ 3 ((or) deg y ≥ 3 and deg z = 2).If z is a cut vertex of G, then let G 1 be a component of G − z with the vertex x not in G 1 .Let z ′ be a vertex farthest away from z in G 1 .
is an independent set of G and S ′ = V(G) − S is an S -fixed connected geodetic set of G and so cg i f (G) ≤ p − 3, but this leads to a contradiction.Hence t = 2. Now claim that each component G i (1 ≤ i ≤ 2) has atleast two vertices.If G 1 has exactly one vertex, say v, then S = {v, z}, where z ∈ G 2 , is an independent set of G.It is clear that S ′ = V(G) − {v, z, u 1 } is an S -fixed connected geodetic set of G and so cg i f (G) ≤ p − 3, but this leads to a contradiction.Hence G has exactly one cut vertex and two components with each component having atleast two vertices.Hence G = K m ← P r → K n (m, n ≥ 2 and r = 1).Case (2) G has two or more cut vertices.Let R = {z 1 , z 2 , . . ., z l }(l ≥ 0) be the set of all cut vertices of degree ≥ 3 in G.If l = 0, then G is a path and so by Theorem 6(i), cg i f (G) = 1 < p − 3, but this leads to a contradiction.Now, let l ≥ 1 and let S = {x 1 , x 2 , . . ., x b } (b ≥ max{2, l}), where x i is a simplicial vertex of G in the i th component of G − R. If l ≥ 3, then clearly S is an independent set of G and S ′ = V(G) − S is an S -fixed connected geodetic set of G and so cg i f (G) ≤ p − 3, but this leads to a contradiction.If l = 1, then clearly S is an independent set of G and S ′ = V(G) − {S ∪ (Q − R)} is an S -fixed connected geodetic set of G. Hence cg i f (G) ≤ p − 3, but this leads to a contradiction.Hence l = 2.If G − R has 3 or more components, then S is an independent set of G and S ′ = V(G) − S is an S -fixed connected geodetic set of G and so cg i f (G) ≤ p − 3, but this leads to a contradiction.Hence G − R has exactly 2 components.Since every vertex of G is either a cut vertex or a simplicial vertex, the components of G − R are complete graphs.Since l = 2, the two components of G − R are complete graphs with atleast two vertices and ⟨R⟩ is a path.Hence G = K m ← P r → K n (m, n ≥ 2 and r ≥ 1).

Theorem 9 .
For any three positive integers a, b and p with 2 ≤ a ≤ b ≤ p − 3, it is possible to identify a connected graph G of order p with g i f (G) = a and cg i f (G) = b.Proof.We consider two cases.Case (1) 2 ≤ a = b ≤ p − 3. Let P p−a−1 : v 1 , v 2 , .. ., v p−a−1 be a path of order p − a − 1 and let K a+1 be the complete graph of order a + 1.Let G be the graph obtained from P p−a−1 and K a+1 by joining an end vertex v p−a−1 of P p−a−1 with every vertex of K a+1 .The resultant graph G is shown in Figure4and its order is p.
Case (2) 2 ≤ a < b ≤ p − 3. Let P p−a−2 : v 1 , v 2 , . . ., v p−b−2 , . . ., v p−a−2 be a path of order p − a − 2. Let K 2 and K a be two complete graphs of orders 2 and a, respectively.Let G be the graph obtained from P p−a−2 , K 2 and K a , by joining the vertex v p−b−1 of P p−a−2 with every vertex of K 2 and joining the end vertex v p−a−2 of P p−a−2 with every vertex of K a .The resultant graph G is shown in Figure 5 and its order is p.

Theorem 10 .
For any three positive integers r, d and n with r ≤ d ≤ 2r, a connected graph G can be identified with radius r, diameter d and the independent fixed connected geodetic number n. Proof.If r = 1, then d = 1 or 2. If d = 1, then by Theorem 7, G = K n+1 has the desired property.Now, let d = 2. Let G be the graph obtained from the complete graphs K 2 and K n+2 by merging a vertex of K 2 , say y, and a vertex of K n+2 .Then G has radius 1, diameter 2 and is shown in Figure 6.