Further new results on strong resolving partitions for graphs

Remark 1

1. If ∂(G) = ε(G), then G _SR ≅ K _|∂(G)|. In particular, (K _n )_SR ≅ K _n and for any tree T with l(T) leaves, (T)_SR ≅ K _l(T).

2. For any connected block graph3 ^[3] G of order n and c cut vertices, G _SR ≅ K _n−c .

3. For any 2-antipodal graph G of order n, G SR ≅ ∪ i = 1 n 2 K 2 . In particular, ( C 2 k ) SR ≅ ∪ i = 1 k K 2 and for any hypercube graph Q _r , ( Q r ) SR ≅ ∪ i = 1 2 r − 1 K 2 .

4. For any positive integer k, (C _2k+1)_SR ≅ C _2k+1.

5. For any positive integers r, t, (K _r,t )_SR ≅ K _r ∪ K _t .

Theorem 2

[20] For any connected graph G, pd_s(G) ≤ dim _s (G) + 1.

Theorem 3

[20] For any connected graph G, pd_s(G) ≤ α(G _SR) + 1.

Theorem 4

[20] For any connected graph G, pd_s(G) ≥ ω(G _SR).

Problem.

Is it true that dim s ( G ) ≤ pd s ( G ) + n − 2 2 for every nontrivial connected graph G of order n?

Proposition 5

For any three integers a, b, c such that a ≥ 0 and b ≤ c,

Proof

Let B = {u ₁,…,u _b } and C = {v ₁,…,v _c } be the bipartition sets of the complete bipartite graph K _b,c used to generate G _a,b,c , and let P _a = w ₁, w ₂,…,w _a if a > 0. Assume that the edge w _a v _c has been added to have G _a,b,c . We observe that any two vertices of B are mutually maximally distant between them and they are not mutually maximally distant with any other vertex of G _a,b,c . Moreover, any two vertices of C − {v _c } ∪ {w ₁} if a > 0, or any two vertices of C if a = 0, are mutually maximally distant between them and they are not mutually maximally distant with any other vertex of G _a,b,c . Thus, in any case (G _a,b,c )_SR ≅ K _b ∪ K _c . By Theorem 4 and Remark 1, we obtain that pd_s(G _a,b,c ) ≥ ω(K _b ∪ K _c ) = c, and by equality (1), dim _s (G _a,b,c ) = b + c − 2.□

Theorem 6

For any integers r, t, n such that 2 ≤ r ≤ t ≤ 2r − 3, there exists a connected graph G of order n with pd_s(G) = r and dim _s (G) = t.

Proof

To obtain our result, we just need to consider a graph G _a,b,c as described above, where a = n − t − 2, b = t − r + 2 (with b < c) and c = r. Clearly, G _a,b,c has the order a + b + c = n − t − 2 + t − r + 2 + r = n and satisfies dim _s (G _a,b,c ) = b + c − 2 = t − r + 2 + r − 2 = t and pd_s(G _a,b,c ) = c = r. Since a ≥ 0 and 1 ≤ b < c, we obtain that t ≤ n − 2, r ≤ t − 1 and t ≤ 2r − 3, which coincides with the condition that n, pd_s(G) and dim _s (G) satisfy.□

Problem.

Is it true that dim _s (G) ≤ 2pd_s(G) − 3 for every nontrivial connected graph G?

We close this section by noticing that the case a = 0 in Proposition 5 leads to the complete bipartite graph K _b,c . In this sense, the following result can be deduced from Proposition 5.

Corollary 7

For any integers r, t with 1 ≤ r ≤ t,

Claim 8.

1. Any two distinct vertices from two different copies of H in any corona graph G ⊙ H belong to different sets in any strong resolving partition for G ⊙ H .

2. Any two distinct vertices from the same set of a strong resolving partition for a corona graph G ⊙ H belong to the same copy of the graph H.

3. Any two distinct vertices from a set A ⊂ V(H _i ) of a strong resolving partition for a corona graph G ⊙ H are strongly resolved by a set B which is a subset of the set V(H _i ).

Proof

(i) Follows from the fact that any two distinct vertices from two different copies of H are mutually maximally distant between them, and thus they are not resolved by any vertex other than themselves. Consequently, any two distinct vertices from the same set of a strong resolving partition for G ⊙ H must belong to the same copy of the graph H, which is the statement of (ii). Finally, (iii) follows since any two vertices of a same copy of H have the same distance to any other vertex of G ⊙ H which is not in this copy.□

Corollary 9

For any connected graph G of order n and any connected graph H, pd s ( G ⊙ H ) ≥ n pd s ( H ) .

Proof

Let Π be a strong partition basis for G ⊙ H . For any i ∈ {1,…,n}, the restriction Π _i of Π to the copy H _i of H in G ⊙ H is a strong resolving partition for H _i . Since every set in Π _i has empty intersection with V(H _j ) for every j ≠ i, it follows that

and the proof is completed.□

Theorem 10

Let G be a connected graph of order n. If H is a complete graph K _m or an edge-less graph N _m , then pd s ( G ⊙ H ) = n m .

Proof

It is easy to see that any two distinct vertices of any copy of H (including the case in which the vertices are in the same copy) are mutually maximally distant between them. Thus ( G ⊙ H ) SR ≅ K n m . Therefore, the result follows from Theorems 3 and 4.□

Theorem 11

Let G be a connected graph of order n. If H has diameter 2, then pd s ( G ⊙ H ) = n pd s ( H ) .

Proof

Let Π _i = {A _i,1,…,A _i,r } be a strong resolving partition for H _i with i ∈ {1,…,n} and r = pd_s(H). We shall prove that the vertex partition Π = {A _1,1 ∪ {v ₁}, A _1,2,…,A _1,r , A _2,1 ∪ {v ₂}, A _2,2,…,A _2,r ,…,A _n,1 ∪ {v _n }, A _n,2,…,A _n,r } is a strong resolving partition for G ⊙ H .

Let x, y be two distinct vertices of G ⊙ H belonging to the same set of Π. If x = v _j for some j ∈ {1,…,n}, then y ∈ A _j,ℓ, for some ℓ ∈ {1,…,r}, and they are adjacent. Hence, for any B ∈ Π such that B ∩ V(H _j ) = ∅, it clearly follows that d G ⊙ H ( y , B ) = d G ⊙ H ( y , x ) + d G ⊙ H ( x , B ) . Thus, x, y are strongly resolved by B. On the other hand, if x, y ∈ A _j,k for some j ∈ {1,…,n} and k ∈ {1,…,r}, then they are strongly resolved in H _j by some set A _j,l with l ≠ k, which means that without loss of generality, d _{H j} (x, A _j,l ) = d _{H j} (x, y) + d _{H j} (y, A _j,l ). Since H has diameter 2, d G ⊙ H ( a , b ) = d H j ( a , b ) for any a, b ∈ V(H _j ), and we have that

Thus, x, y are strongly resolved by A _j,l . As a consequence, Π is a strong resolving partition for G ⊙ H and we obtain that pd s ( G ⊙ H ) ≤ n pd s ( H ) . The equality is finally obtained by using Corollary 9.□

Theorem 12

For any connected graph G of order n and any graph H of diameter at least3, pd s ( G ⊙ H ) = n pd s ( K 1 + H ) .

Proof

By using a similar procedure like that in the proof of the upper bound in Theorem 11, we obtain the upper bound pd s ( G ⊙ H ) ≤ n pd s ( K 1 + H ) . Notice that now, the given partition is obtained from a strong resolving partition for K 1 ⊙ H instead of that for H (like in Theorem 11).□

Remark 13

Let G be any connected graph of diameter at least 3.

1. If G has true twins, then (K ₁ + G)_SR ≅ G ^ct .

2. If G has no true twins, then (K ₁ + G)_SR ≅ G ^c .

A set S is an independent set in a graph G, if the subgraph induced by S has no edges. The maximum cardinality of any independent set is the independence number of G and is denoted by β(G). Now, by using the aforementioned remark we derive some conclusions on the strong partition dimension of the join graph K ₁ + G for some classes of graphs G.

Proposition 14

Let G be any connected graph of order n and diameter at least 3.

1. If G has t true twins, then β(G) ≤ pd_s(K ₁ + G) ≤ n − ω(G) + t.

2. If G has no true twins, then β(G) ≤ pd_s(K ₁ + G) ≤ n − ω(G) + 1.

Proof

The lower bounds follow from Theorem 4 and Remark 13. That is pd_s(K ₁ + G) ≥ ω((K ₁ + G)_SR) = ω(G′) ≥ ω(G ^c ) = β(G), where G′ is either G ^c or G ^ct .

To prove the upper bounds, we use Theorem 3 and Remark 13. For the case of (i), let G* be the subgraph of G ^ct induced by the true twin vertices of G. Thus, we have

and for the case (ii) it follows:

which completes the proof.□

Proposition 15

1. For any cycle graph C _n with n ≥ 6, pd s ( K 1 + C n ) = ⌈ n 2 ⌉ .

2. For any path graph P _n with n ≥ 5, pd s ( K 1 + P n ) = ⌈ n 2 ⌉ .

Proof

(i) We observe that for any three distinct vertices of the cycle C _n , at least two of them are at distance 2 and they are mutually maximally distant, since K ₁ + C _n has diameter 2. Thus, any set belonging to any strong resolving partition for K ₁ + C _n contains at most two vertices of the cycle C _n . As a consequence, we obtain that p d s ( K 1 + C n ) ≥ ⌈ n 2 ⌉ . On the other hand, let u be the vertex of K ₁ and let v ₁, v ₂,…,v _n be the vertex set of C _n . Hence, it is not difficult to observe that the partition Π = {{u, v ₁, v ₂}, {v ₃, v ₄},…,{v _n−1, v _n }}, if n is even, or Π = {{u, v ₁, v ₂}, {v ₃,v ₄},…,{v _n−2, v _n−1}, {v _n }}, if n is odd, is a strong resolving partition for K ₁ + C _n . Thus, the equality follows since | Π | = ⌈ n 2 ⌉ .

(ii) A similar argument as above gives the result.□

Theorem 16

[25] For any circulant graph CR(n, t) with 0 < t < ⌊ n 2 ⌋ , pd(CR(n, t)) ≥ t + 1.

By using inequality (2) and Theorem 16, we are able to give the following bounds for the strong partition dimension of circulant graphs.

Theorem 17

For any circulant graph CR(n,t) with 0 < t < ⌊ n 2 ⌋ ,

Proof

The lower bound follows by using inequality (2) and Theorem 16. For the upper bound, let V = {v ₀,v ₁,…,v _n−1} be the vertex set of CR(n,t). From now on, in this proof, operations with the subindices of vertices in V are done modulo n. Assume that every vertex v _i is adjacent to v _i+j for every j ∈ {−t, −t + 1,…,−1, 1,…,t}. We shall prove that the partition Π = {A ₀, A ₁,…,A _t+1} such that A ₀ = {v ₀}, A t = { v t , v t + 1 , … , v t + ⌈ ( n − t ) / 2 ⌉ } and A t + 1 = { v t + ⌈ ( n − t ) / 2 ⌉ + 1 , v t + ⌈ ( n − t ) / 2 ⌉ + 2 , … , v n − 1 } is a strong resolving partition for CR(n, t).

Proposition 18

For any bouquet of cycles B ∈ ℬ a , b , c ,

Proof

According to the construction of the strong resolving graph B _SR, by using Theorem 4, the lower bound follows from the fact that the set V _a ∪ V _2c together with exactly one vertex from each bipartition set of the subgraph induced by V _2b forms a clique in B _SR.

On the other hand, we consider the following vertex partition, denoted by Π, of V(B).

• For every even cycle C _{r i} , i ∈ {1,…,a}, we create a set A _i = {v}, where v is the vertex diametral with w in C _{r i} .

• For every odd cycle C _{s i} , i ∈ {1,…,b}, we create a set B _i = {u, v}, where u, v are the vertices diametral with w in C _{s i} .

• For every cycle C 3 j of order three, we create two sets D _j and F _j , each of them containing one of the two vertices different from w.

• Finally, we create two sets R and S in the following way. If the vertex set of a cycle C _n (even or odd, but not of order three) of B is represented as V(C _n ) = {w = w ₀, w ₁,…,w _n−1} (with the natural style of adjacency between the vertices), then R ∩ V ( C n ) = { w 0 , w 1 , … , w ⌊ n / 2 ⌋ − 1 } and S ∩ V ( C n ) = { w ⌈ n / 2 ⌉ + 1 , w ⌈ n / 2 ⌉ + 2 , … , w n − 1 } .