Corrigendum to “On the monophonic rank of a graph” [Discrete Math. Theor. Comput. Sci. 24:2 (2022) #3]

In this corrigendum


The counterexample
In the paper "On the monophonic rank of a graph", which appeared in Discrete Math.Theor.Comput.Sci.24:2 (2022) #3, we claimed in Theorem 5.2 the NP-completeness of MONOPHONIC RANK for 2-starlike graphs.This result was used in Corollary 5.1 to prove the NP-completeness of MONOPHONIC RANK for k-starlike graphs for any fixed k ≥ 2. However, the reduction given in Theorem 5.2 is not correct.We show in the sequel that the example given in Figure 1 illustrates a counterexample.Consequently, Corollary 5.1 does not hold as well.
We need some definitions.Consider a graph G.The open and the closed neighborhoods of a vertex v are denoted by N (v) and N [v], respectively.A vertex is simplicial if its closed neighborhood induces a complete graph.It is clear that a simplicial vertex is not an internal vertex of an induced path.We say that S is monophonically convex if the vertices of every induced path joining two vertices of S are contained in S. The monophonic convex hull of S, S , is the smallest monophonically convex set containing S. A set S is monophonic convexly independent if v is not in S − {v} for v ∈ S. The monophonic rank of G, r(G), is the size of a largest monophonic convexly independent set of G.
Recall that a starlike graph G admits a partition of (Gustedt (1993); Cerioli and Szwarcfiter (2006)).The 1-starlike 2 Mitre Dourado et.al w 14 Corrigendum to "On the monophonic rank of a graph" 3 graphs are the split graphs.Denote by Z the set of simplicial vertices of G.For i ∈ {0, 1, . . ., t}, we denote by X i the maximal clique containing . ., t}, and that C 0 can be an empty set.
Lemma 1.1 If G is a starlike graph and S ⊆ V (G), then every vertex v ∈ S − S belongs to C ′ 0 and is an internal vertex of an induced (u, u ′ )-path such that u ∈ S ∩ Z ∩ N (v) and u ′ ∈ S .
Proof: Let v ∈ S −S.It is clear that v is an internal vertex of an induced (w, w ′ )-path P for w, w ′ ∈ S .Since Z contains only simplicial vertices, we have that v ∈ C ′ 0 .Since C ′ 0 is a clique, we have that at least one of w and w ′ , say w, belongs to Z. Suppose first that w ′ also belongs to Z.Note that P has 3 or 4 vertices.In both cases, v is adjacent to at least one of w and w ′ .Suppose then that w ′ belongs to C ′ 0 .In this case, P has 3 vertices, which means that v is adjacent to w.In all cases, v is adjacent to some vertex of Z belonging to S .Since every vertex of Z is simplicial, we have that such vertex belongs to S, completing the proof. ✷ Figure 1 shows an input graph G with n = 7 vertices and the resulting 2-starlike graph G ′ according to the reduction given in Theorem 5.2 in (Dourado et al. (2022)), whose vertex set can be partitioned into sets U and W where U is a clique with 70 vertices and G[W ] has maximum clique of size 2. In such proof, we claimed that G has an independent set with ⌈ n+1 2 ⌉ = 4 vertices if and only if G ′ has a monophonic convexly independent set with p vertices, where p = n + (4n − 1)⌈ n+1 2 ⌉ = 7 + 27 × 4 = 115.It is easy to see that G has no independent set of size ⌈ n+1 2 ⌉ = 4.In order to see that the set S ⊂ V (G ′ ) with 115 vertices formed by the 56 vertices of U 1 , U 2 , W 1 , W 2 , the 56 vertices of U 4 , U 5 , U 6 , U 7 and the 3 vertices u 15 5 , u 15 6 and u 15 7 is m-convexly independent we use the notation given above where 2 Monophonic rank is polynomial for starlike graphs In Theorem 5.1 of (Dourado et al. (2022)), we presented a polynomial-time algorithm for computing the monophonic rank of 1-starlike graphs.Here, in Corollary 2.1, we extend such algorithm so that it works for starlike graphs.
Given a graph G and a set T ⊆ V (G), we write N (T ) = ∪ Using the notation given previously for a starlike graph G with partition Note that G i is a split graph where Y i is an independent set which can be empty.If this is the case, G i can be the empty graph.See an example in Figure 2.
Lemma 2.1 If G is a starlike graph, then for i ∈ {0, 1, . . ., t}, G i has a critical independent set Fig. 2: Graphs G0, G1 and G3 constructed from the starlike graph G. Ellipses formed by continuous lines represent cliques, while the ones formed by dashed lines represent independent sets.

v∈TN
(v).If T is an independent set, we define the difference of T as d(T ) = |T | − |N (T )|, and the critical independence difference of G as d c (G) = max{d(T ) : T is an independent set of G}.If d(T ) = d c (G), then we say that T is a critical independent set of G.