In recent years, there has been an increasing interest in connected p-center and p-median problems where the subgraph induced by the selected set is connected. Yen [1] studied the connected p-center problem on block graphs. Bai et al. [2] considered the connected p-median problem on cactus graphs and showed that the problem can be solved in polynomial time. In this paper we consider the inverse connected p-median problem on block graphs. We shall follow the notations and terminologies given in Kang et al. [3] , Nguyen and Hung [4] . Let G = ( V , E , w , l ) be a finite, connected, undirected graph with vertex set V of order n = | V | and edge set E of size m = | E | , where each vertex v ∈ V is associated with a nonnegative weight w ( v ) and each edge ( v i , v j ) ∈ E is associated with a certain cost or length l ( v i , v j ) . For convenience, we denote G = ( V , E , w ) as the graph that l ( e ) = 1 for all edge e ∈ E .

For any two vertices of u , v ∈ G , a path from u to v is vertex-edge alternative sequence: u = x 1 , e 1 , x 2 , e 2 , ⋯ , x s , e s , x s + 1 = v such that the x i are all distinct and e i = x i x i + 1 for i = 1 , 2 , ⋯ , s . The number of edges of a path is its length. Let d ( u , v ) = ∑ i = 1 s     l ( e i ) be the length of a shortest path in G between u and v, called the distance of two vertices u and v. Furthermore, each edge e = ( u , v ) can be considered as a continuous interval, where a point ρ in e is identified by a parameter λ ∈ [ 0,1 ] such that d ( u , ρ ) = λ l ( e ) and d ( v , ρ ) = ( 1 − ρ ) l ( e ) . We can also define the distance between two points similarly to the distance between two vertices. The classical p-median location model is to find the set of p points on G, say S = { ρ 1 , ρ 2 , ⋯ , ρ p } , so as to minimize the median function

F ( S ) = ∑ v ∈ V     w ( v ) d ( v , S ) ,

where d ( v , S ) = min j = 1 p d ( v , ρ j ) . By the dominating property of vertex set, we know that there exists an optimal solution to the p-median problem that is exactly the subset of V. Hence, we focus on the set S = { v 1 , v 2 , ⋯ , v p } ⊂ V hereafter. For the sake of modern location model, the new facilities are required to be connected to a network for communication/security reasons. This fact motivates the so-called connected p-median problem on G, where the set S is the connected set on the underlying graph. For two vertices set S = { v 1 , v 2 , ⋯ , v p } and S ′ = { u 1 , u 2 , ⋯ , u q } , we define d ( S , S ′ ) = min { d ( v i , u j ) | v i ∈ S , u j ∈ S ′ } .

Given a graph G, a vertex u is called a cut vertex of G if κ ( G − { u } ) > κ ( G ) , where κ ( G ) denotes the number of components of G. A connected subgraph H of G is called a block of G if H is maximal and it contains no cut vertices. A graph G is a block graph if all blocks of G are cliques and any two distinct blocks B 1 and B 2 have at most one common vertex, where a clique in a graph is a complete subgraph maximal under inclusion.

Given a block graph G and a set S p of p connected vertices, we denote by B ( S p ) the set of connected subgraphs induced by deleting all vertices in S p and all edges in blocks containing at least one vertex of S p . A vertex v is said to be in the border of S p if there does not exist any pair v ′ and v ″ in S p such that v is an intermediate vertex of the shortest path connecting them. Resultantly, we denote D ( S p ) the set of all vertices in the border of S p . Furthermore, let F ( S p ) be the set of vertices, which are adjacent to some vertices in D ( S p ) and are not in S p . Meanwhile, we define F ( S p ) ( v ) , for v in D ( S p ) , as the set of all vertices in F ( S p ) that is adjacent to v. If a connected subgraph in B ( S p ) contains a vertex v ∈ F ( S p ) , then this graph is defined as B ( S p ) ( v ) . For a vertex v ∈ D ( S p ) , let B ¯ ( S p ) ( v ) be the subgraph induced by { v } ∪ ∪ v ′ ∈ F ( S p ) ( v )     B ( S p ) ( v ′ ) .

Modifying vertex weights of a graph at minimum total cost so that a predetermined set of p connected vertices becomes a connected p-median on the perturbed graph. This problem is the so-called inverse connected p-median problem on graphs. Nguyen and Hung [4] consider the problem of a block graph with uniform edge lengths under various cost functions. To solve the problem, they first find an optimality criterion for a set that is a connected p-median. Based on this criterion, they can formulate the problem as a convex or quasiconvex univariate optimization problem. Finally, they develop combinatorial algorithms that solve the problems under the three cost functions in O ( n log n ) time, where n is the number of vertices in the underlying block graph. In the scope of their paper, the following cost functions are considered.

1) Rectilinear norm:

C ( p v , q v ) = ∑ v ∈ V ( c v p v + c v q v ) .

2) Chebyshev norm:

C ( p v , q v ) = max v ∈ V { c v p v , c v q v } .

3) Bottleneck Hamming distance:

C ( p v , q v ) = max v ∈ V { c v H ( p v ) , c v H ( q v ) } ,

where the Hamming distance H ( ⋅ ) is identified by H ( x ) = 0 if x = 0 and H ( x ) = 1 otherwise. The following results are established in Nguyen and Hung [4] .

Lemma 1. [4] If there exists a vertex v ∈ D ( S p ) and vertex u ∈ F ( S p ) such that W ( B ( S p ) ( u ) ) > W ( B ¯ ( S p ) ( v ) ) , then S p is not a connected p-median of G.

Theorem 1. [4] (Optimality Criterion) The set S p is a connected p-median of the block graph G if and only if W ( B ( S p ) ( u ) ) ≤ W ( B ¯ ( S p ) ( v ) ) for all v ∈ D ( S p ) and u ∈ F ( S p ) .

Theorem 2. [4] The inverse connected p-median problem on a block graph can be solved in O ( n log n ) time.

Theorem 3. [4] The inverse connected p-median problem on block graphs under Chebyshev norm can be solved in O ( n log n ) time.

Theorem 4. [4] The inverse connected p-median problem under bottleneck Hamming distance can be solved in O ( n log n ) time.

As we observe, Lemma 1 and Theorem 1 in Nguyen and Hung [4] are incorrect. In this note, we point out these incorrect results by a counterexample.

Counterexample 1. Let us consider the block graph G in Figure 1, where the prespecified set of connected vertices is S 4 = { v 1 , v 2 , v 3 , v 4 } and the weights are labeled on the corresponding vertices.

According to the symbols and definitions in Nguyen and Hung [4] , one can deduce the following results. D ( S 4 ) = { v 1 , v 2 , v 4 } , F ( S 4 ) = { v 5 , v 6 , v 9 , v 10 , v 11 , v 13 } . Moreover, the subgraph B ( S 4 ) ( v 13 ) is induced by { v 11 , v 12 , v 13 , v 14 } and the subgraph B ¯ ( S 4 ) ( v 4 ) is induced by { v 4 , v 11 , v 12 , v 13 , v 14 } . Based on the weights of vertices, one obtains the weights of subgraphs are calculated as in Table 1.

From Table 1, one can see that W ( B ( S p ) ( v j ) ) ≤ W ( B ¯ ( S p ) ( v i ) ) for all v i ∈ D ( S p ) and v j ∈ F ( S p ) . If Lemma 1 and Theorem 1 are correct, then the set S 4 is a connected p-median of the block graph. Let S ′ 4 = { v 7 , v 2 , v 3 , v 4 } . Since F ( S ′ 4 ) = 20 < F ( S 4 ) = 23 , the set S 4 is not a connected p-median of the block graph, this instance is indeed a counterexample for Lemma 1 and Theorem 1.

In addition, the proofs in Lemma 1 and Theorem 1 in Nguyen and Hung [4] are also incorrect. Let S ″ 4 = { v 11 , v 2 , v 3 , v 4 } . According to the symbols and definitions in Nguyen and Hung [4] , one can deduce that the subgraph B ( S 4 ) ( v 11 ) is induced by { v 11 , v 12 , v 13 , v 14 } and the subgraph B ¯ ( S 4 ) ( v 1 ) is induced by { v 4 , v 5 } . Then

W ( B ( S 4 ) ( v 11 ) ) = 4     and     W ( B ¯ ( S 4 ) ( v 1 ) ) = 4.

Since

F ( S ″ 4 ) = 25 ≠ F ( S 4 ) + W ( B ¯ ( S 4 ) ( v 1 ) ) − W ( B ( S 4 ) ( v 11 ) ) = 23,

the deduction in Lemma 1 and Theorem 1 are clearly incorrect.

The Lemma 1 and Theorem 1 in Nguyen and Hung [4] are incorrect mainly because of the following notation definitions: B ( S p ) , D ( S p ) , F ( S p ) , F ( S p ) ( v ) , B ( S p ) ( v ) and B ¯ ( S p ) ( v ) . In this article, we will preserve the symbol definition of B ( S p ) and D ( S p ) , redefine symbols: B ( S p ) ( v ) and B ¯ ( S p ) ( v ) , and do not use F ( S p ) and F ( S p ) ( v ) anymore to avoid confusion with F ( S p ) .

For a vertex v ∈ D ( S p ) , let B ¯ ( S p ) ( v ) be the subgraph induced by vertices that can only be served by v. In other words, for vertices v i ∈ B ¯ ( S p ) ( v ) , only vertex v ∈ D ( S p ) satisfies d ( v i , v ) = d ( v i , S p ) . Let N ( S p ) be the set of vertices of G − S p , which are adjacent to some vertices in S p . If the shortest path from v i ∈ B ( S p ) to S p passes through u ∈ N ( S p ) , then v i is said to be served by S p through u. For a vertex u ∈ N ( S p ) , let B ( S p ) ( u ) be the subgraph induced by the vertices which are served by S p through u.

Example 1. Let us consider the block graph G in Figure 1, where the prespecified set of connected vertices is S 4 = { v 1 , v 2 , v 3 , v 4 } and the weights are labeled on the corresponding vertices.

According to the new symbols and definitions above, one can deduce the following results. D ( S 4 ) = { v 1 , v 2 , v 4 } , N ( S 4 ) = { v 5 , v 6 , v 7 , v 9 , v 10 , v 11 , v 13 } . Moreover, the subgraph B ¯ ( S 4 ) ( v 3 ) is induced by { v 3 , v 7 , v 8 } and the subgraph B ( S 4 ) ( v 13 ) is induced by { v 13 , v 14 } .

Using the same method in the proof of Lemma 1 in Nguyen and Hung [4] , the following lemma can be proved.

Lemma 2. If there exists a vertex v ∈ D ( S p ) and vertex u ∈ N ( S p ) such that W ( B ( S p ) ( u ) ) > W ( B ¯ ( S p ) ( v ) ) , then S p is not a connected p-median of G ( V , E , w ) .

There is no problem in the method and thought of proving Theorem 1 in Nguyen and Hung [4] , but the description is not clear and concise enough. We will give a new proof of Theorem 1 in the following.

Theorem 5. (Optimality Criterion) The set S p is a connected p-median of the block graph G ( V , E , w ) if and only if W ( B ( S p ) ( u ) ) ≤ W ( B ¯ ( S p ) ( v ) ) for all v ∈ D ( S p ) and u ∈ N ( S p ) .

Proof. If S p is a connected p-median of the block graph G, then W ( B ( S p ) ( u ) ) ≤ W ( B ¯ ( S p ) ( v ) ) for all v ∈ D ( S p ) and u ∈ N ( S p ) . This is the converse-negative proposition of Lemma 2, which is clearly true.

Conversely, we prove that if W ( B ( S p ) ( u ) ) ≤ W ( B ¯ ( S p ) ( v ) ) for all v ∈ D ( S p ) and u ∈ N ( S p ) , then the set S p is a connected p-median of the block graph G. Let S p * be a connected p-median such that S p * ≠ S p and d ( S p , S p * ) is as small as possible. Furthermore, if d ( S p , S p * ) = 0 , let’s assume that S p * ∩ S p contains as many vertices as possible. We take a vertex v ′ ∈ D ( S p * ) such that v ′ ∉ S p . As ∪ u ∈ N ( S p )     B ( S p ) ( u ) = V \ S p , we know that there exists a vertex u ″ ∈ N ( S p ) such that v ′ ∈ B ( S p ) ( u ″ ) . Hence W ( B ¯ ( S p * ) ( v ′ ) ) ≤ W ( B ( S p ) ( u ″ ) ) . Similarly, we choose a vertex v ″ ∈ D ( S p ) such that v ″ ∉ S p * . There exists a vertex u ′ ∈ N ( S p * ) such that v ″ ∈ B ( S p * ) ( u ′ ) . Hence W ( B ¯ ( S p ) ( v ″ ) ) ≤ W ( B ( S p * ) ( u ′ ) ) . By assumption of S p , W ( B ( S p ) ( u ″ ) ) ≤ W ( B ¯ ( S p ) ( v ″ ) ) . Hence, W ( B ¯ ( S p * ) ( v ′ ) ) ≤ W ( B ( S p * ) ( u ′ ) ) . On the other hand, since S p * is a connected p-median, W ( B ¯ ( S p * ) ( v ′ ) ) ≥ W ( B ( S p * ) ( u ′ ) ) . Hence, W ( B ¯ ( S p * ) ( v ′ ) ) = W ( B ( S p * ) ( u ′ ) ) . We set S ′ p = ( S p * \ { v ′ } ) ∪ { u ′ } . Obviously S ′ p is connected. Using the same method in the proof of Lemma 1 in Nguyen and Hung [4] , we have F ( S ′ p ) = F ( S p * ) . Then S ′ p is also a connected p-median. If d ( S p , S p * ) > 0 , then d ( S p , S ′ p ) < d ( S p , S p * ) . If d ( S p , S p * ) = 0 , then S p ∩ S ′ p contains more vertices than S p ∩ S p * . This contradicts the choice of S p * . Therefore, S p = S p * and S p is a connected p-median of the block graph G.

In the solution approach of the inverse connected p-median problem on block graphs in [4] , we only need to change F ( S p ) , B ( S p ) ( v ) and B ¯ ( S p ) ( v ) into the new definition of N ( S p ) , B ( S p ) ( v ) and B ¯ ( S p ) ( v ) , respectively. The inverse connected p-median problem on block graphs G ( V , E , w ) under various cost functions, say rectilinear norm, Chebyshev norm, and bottleneck Hamming distance, can be solved by using the approach in [4] . However, the description of Theorem 2, Theorem 3 and Theorem 4 in Nguyen and Hung [4] is not complete and rigorous enough. Let’s redescribe the three theorems in the following.

Theorem 6. The inverse connected p-median problem on block graphs G ( V , E , w ) under Rectilinear norm can be solved in O ( n log n ) time, where n is the number of vertices in the block graph.

Theorem 7. The inverse connected p-median problem on block graphs G ( V , E , w ) under Chebyshev norm can be solved in O ( n log n ) time, where n is the number of vertices in the block graph.

Theorem 8. The inverse connected p-median problem on block graphs G ( V , E , w ) under Bottleneck Hamming distance can be solved in O ( n log n ) time, where n is the number of vertices in the block graph.

This work was partially supported by the Natural Science Research Foundation of Colleges and Universities of Anhui Province (KJ2021A0968, KJ2021A0967), the Key Scientific and Technological Project of Higher Education of Henan Province (20A110018), the PhD Research Foundation of Henan Agricultural University (30601668).

The authors declare no conflicts of interest regarding the publication of this paper.

Bai, C.S., Zhang, L.Q. and Zhou, J.J. (2023) A Note on the Inverse Connected p-Median Problem on Block Graphs. Advances in Pure Mathematics, 13, 181-186. https://doi.org/10.4236/apm.2023.134011