The dominating partition dimension and locating-chromatic number of graphs

For every graph G , the dominating partition dimension of G is either the same as its partition dimension or one higher than its partition dimension. In this paper, we consider some general connections among these three graph parameters: partition dimension, locating-chromatic number, and dominating partition dimension. We will show that β p ( G ) ≤ η p ( G ) ≤ χ L ( G ) for any graph G with at least 3 vertices. Therefore, we will derive properties for which graphs G have η p ( G ) = β p ( G ) or η p ( G ) = β p ( G ) + 1 .


Introduction
The study of determining all coordinates of vertices in a graph has received extensive attention.For example, Ore [8] defined dominating sets to determine the location of vertices in a graph by considering the neighbors of each member of a set of vertices.Slater [9] defined the locating set and locating number by assigning a unique distance coordinate for each vertex to a certain subset of vertices.On the other hand, independently, Harary and Melter [6] also studied the same concept but used different terms, namely resolving set and metric dimension.Chartrand, Salehi, and Zhang [5] introduced the resolving partition and partition dimension of a graph as a new perspective in determining the metric dimensions of graphs.
One of the studies to develop the concept of determining the coordinates of vertices in a graph is by combining two well-known concepts.Slater [10] has proposed the idea of combining the resolving set and the dominating set.Later, this study was enhanced by Brigham,et. al. [3] in 2003 and separately by Henning and Oellermann in 2004.Previously, Chartrand et.al. [4] defined a locating-chromatic number, combining the concept of vertex partitioning with vertex coloring.Following the idea of combining the concepts of resolving set and dominating set, Hernando, Mora, and Pelayo [7] expanded the concept of resolving partition by defining dominating partition and dominating partition dimension.They added a dominating condition to a resolving partition of a graph and then called it as the resolving dominating partition of the graph.In this paper, we consider some general connections among these three graph parameters: partition dimension, locating-chromatic number and dominating partition dimension.
Let G be a simple and connected graph with vertex set V (G) and edge set E(G).The distance between vertices u and w in G, denoted by d(u, w), is the length of a shortest path connecting u and w in G.The distance between a vertex u ∈ V (G) and a subset S ⊆ V (G) is the minimum of the distances between u and the vertices of S, that is, The representation of a vertex u ∈ V (G) with respect to Π is the vector consisting all distances from u to all elements of Π, that is, The partition Π is called a resolving dominating partition of G, if it is both resolving partition and dominating partition.A resolving dominating partition of G with minimum cardinality is called a minimum resolving dominating partition of G.The cardinality of a minimum resolving dominating partition of G is called the dominating partition dimension of G, denoted by η p (G).
Let σ be a proper k-coloring of G, which mean that any two adjacent vertices in G have distinct colors.Recall that a proper k-coloring σ is equivalent to a partition where S i is the set of vertices receiving color i for 1 ≤ i ≤ k.Let u ∈ V (G) and Π be a partition of V (G) induced by σ.The color code σ Π (u) of a vertex u is defined as the ordered ktuple (d(u, S 1 ), (u, S 2 ), • • • , (u, S k )).The proper k-coloring σ (or partition Π) is called a locatingchromatic k-coloring of G, locating k-coloring for short, if all vertices of G have distinct color codes.The locating-chromatic number χ L (G) of G is the smallest k such that G has a locating k-coloring, and this locating k-coloring is called a minimum locating coloring of G.

Main Results
For any graph G of order n ≥ 3, Hernando et al. [7] showed the dominating partition dimension of G is equal to either the partition dimension of G or the partition dimension of G plus one.
Theorem 2.1.[7] For any graph G of order n ≥ 3, Based on Theorem 2.1, we can classify all graphs G depending on the value of its dominating partition dimension.A graph G is said to be of type DP1 if η p (G) = β p (G), otherwise we call G as a graph of type DP2 (if η p (G) = β p (G) + 1).
In this paper, we would like to classify which graphs G of type DP1 or type DP2.Before classifying these graphs, let us consider some general connections between these three graph parameters: partition dimension, locating-chromatic number, and dominating partition dimension.
Theorem 2.2.For any connected graph G, every locating coloring of G is also a resolving dominating partition of G.
Proof.Let G be a graph.Let σ be any locating coloring of G and Π := {S 1 , S 2 , • • • , S k } be the partition of V (G) induced by σ.Since σ is a locating coloring of G, then for any two distinct vertices x and y in G there exists S i for some and Π is a resolving partition of G. Now, since every coloring of G is also a proper coloring, then for every two adjacent vertices x, y in G, we have σ(x) ̸ = σ(y).So, this implies that x and y belong to different partition classes of Π.This fact yields that every vertex x is dominated by some partition class S i for some i ∈ [1, k].Thus, Π is a dominating partition of G. Therefore, Π is a resolving dominating partition of G.
The converse of Theorem 2.2 is not always true.The following graph G in Figure 1    There are many classes of graphs with these three parameters having the same values.An example of a graph G with a small order and β p (G) = χ L (G) = η p (G) is given in Figure 3.A resolving partition of G with minimum cardinality is shown in Figure 3  In the following two remaining sections, the graphs which are of type DP1 will be presented in section 3.In Section 4, we will further provide some classes of graphs of type DP2.In particular, we classify some class graphs of type DP1 or type DP2 with small parameters: partition dimension and dominating partition dimension.Before going to the next section, some known results regarding these parameters are shown.

Graphs of type DP1
In this section, we will further derive some graphs G of type DP1, namely graphs G with η p (G) = β p (G).We begin by giving some graphs of type DP1 with a small dominating partition dimension.By Theorem 2.4 point (a) and (b), we have that if , even}, and Now, in the following theorems, we will derive graphs G of type DP1 with η p (G) = 3.This means that G has also β p (G) = 3.To date, there is no complete characterization regarding all graphs with partition dimension three.However, there is a complete characterization on graphs with locating-chromatic three.Let T be a set of all trees T on n vertices (n ≥ 3) with locatingchromatic number three.Baskoro and Asmiati (2013) characterized all the members of such a set T as follows.

Theorem 3.2. [2]
A tree T is in T if and only if T is any subtree of one of the trees (A), (B) or (C) in Figure 4 containing vertices X, Y and Z, with For graphs other than trees, Asmiati and Baskoro [1] have also characterized all such graphs G with χ L (G) = 3.Such graphs are stated in the following theorem.The next corollary gives all graphs of type DP1 with the locating-chromatic number three.Proof.Let G be a graph other than a path with χ L (G) = 3.Then, G must be isomorphic to one of the graphs characterized in Theorem 3.2 or Theorem 3.3.By Corollary 2.1, β p (G) ≤ η p (G) ≤ 3.
Since the only graph with partition dimension two is a path, then β p (G) ≥ 3.This implies that β p (G) = η p (G) = 3.Thus, G is a graph of type DP1 with χ L (G) = 3.

Graphs of type DP2
In this section, we determine graphs on n vertices of type DP2, namely the graphs G with η p (G) = β p (G) + 1.We start this section with a corollary showing that any path with n ≥ 3 vertices is a graph of type DP2.
Proof.Let G be a graph on n ≥ 5 vertices with an earring Since k leaves hanging from x, there are two leaves, w.l.o.g we may assume x 1 and x 2 , to be included in the same partition class of Π.Note that d(x 1 , v) = d(x 2 , v) for all v ∈ V (G)\{x 1 , x 2 }, this implies that r(x 1 |Π) = r(x 2 |Π), a contradiction.Therefore β p (G) ≥ k.Furthermore, let σ be a locating-coloring of G.Each locating-coloring σ of G assigns distinct colors to these k leaves.Since x is adjacent to these leaves, it must be colored a different color than the k colors that have been used on the leaves.Therefore, χ L (G) ≥ k + 1.
Next, we show that Now, if η p (G) = k then there exists a resolving partition Π of cardinality k.Since there are k leaves hanging from x, then each leaf hanging from x is in a different partition class of Π.Thus, there is a partition class of Π that contains both x and a leaf hanging from x.It implies that Π is not dominating partition.Hence, η p (G) ≥ k + 1, and then η p (G) = k + 1.
Let M t+1 be a tree of order t + 1 with 2 ≤ t ≤ n 2 for any integers n and t.Let T n be a tree of order n obtained by connecting n − t − 1 new vertices to a vertex that is not an earring in the M t+1 .Syofyan et al. [11] characterized all trees of order n ≥ 6 with locating-chromatic number n − t.The characterization is as follows.

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The dominating partition dimension and locating-chromatic number of graphs | M. Ridwan et al.

Theorem 4.2.
[11] Let T n be a tree of order n with n ≥ 6.Then, χ L (T n ) = n − t where 2 ≤ t < n 2 if and only if T n has exactly one earring of size n − t − 1.Now, we characterize all trees on n vertices with dominating partition dimension n−t for some t.
Theorem 4.3.Let T n be a tree of order n with n ≥ 6.Then, η p (T n ) = n − t for some 2 ≤ t < n 2 if and only if T n has exactly one earring of size n − t − 1.
Proof.Let T n be a tree of order n with n ≥ 6.If T n has exactly one earring of size n − t − 1 for some fixed 2 ≤ t < n 2 , then by Theorem 4.1,  If G is not a tree and G is of type DP2 with η p (G) = k for some integer k ≥ 3, then G does not necessarily contain an earring of size k − 1.For any integers m, t ≥ 3, let us consider a graph G having one earring of size 2 as depicted in Figure 6.We will show that this graph G is of type DP2 as stated in the following theorem.The number of these graphs G are infinite since m and t can be arbitrary integers greater than or equal to 3.  Proof.Let G be the graph in Figure 6 for some integers m ≥ 3 and t ≥ 3. Notice that, G contains an earring x 1 of size 2 with the vertices w 1 and v 1 as leaves hanging at x 1 .To prove that G is a graph of type DP2, we must show first that β p (G) = 3.Since G is not a path, then by Theorem 2.4 part (a) we have β p (G) ≥ 3. Now, take an ordered partition Γ = {S 1 , S 2 , S 3 } where since otherwise r(w Case 2. c(M ) contains exactly two '1's.There are 3 subcases to be considered: Consider the first subcase, without loss of generality, we my assume c(v 2 ) = 2.Note that, either c( To conclude this proof, we obtain that η p (G) = 4 and β p (G) = 3.Then G is a graph of type DP2.
has η p (G) = β p (G) = 4, but χ L (G) = 5.A resolving partition as well as a resolving dominating partition of G with minimum cardinality is shown in Figure 1(a).A locating coloring of G with a minimum number of colors is shown in Figure 1(b).
(a).A resolving dominating partition of G with minimum cardinality is shown in Figure 3(b).

Theorem 3 . 3 . 1 .
[1] Let G be a graph other than a tree with χ L (G) = 3.Then, If G is bipartite then G is isomorphic to any subgraph of the graph in Figure5(A) containing at least all blue edges.2. If G is not bipartite then G is isomorphic to any subgraph of either the graph (B), (C), (D),or (E) in Figure5containing the smallest odd blue cycle C m .

Figure 4 .
Figure 4.All trees T with χ L (T ) = 3 and with a minimum locating coloring.

Figure 5 .
Figure 5.All graphs G other than trees with χ L (G) = 3 and with a minimum locating coloring.

Corollary 3 . 1 .
Let G be a graph other than a path with χ L (G) = 3.Then G belongs type DP1.

Corollary 4 . 1 .
P n on n ≥ 3 vertices is a graph of type DP2.Proof.From Theorem 2.4 part (a) we have that β p (P n ) = 2.By Theorem 2.4 part (c), we conclude that η p (P n ) = 3 = β p (P n ) + 1.Therefore, P n on n ≥ 3 vertices is a graph of type DP2.

Theorem 4 . 1 .
If G is a graph on n ≥ 5 vertices having a unique earring of size k, with ⌈ be the leaves adjacent to earring x in G, and a 1 , a 2 , • • • , a n−k−1 are the remaining vertices of G. Assume that β p (G) = k − 1 and Π is a resolving partition of G with (k − 1)-classes.
Then, by Theorem 4.2, T n has exactly one earring of size n−t ′ −1.Thus, by Theorem 4.1, η p(T n ) = χ L (T n ) = n−t ′ > n−t,a contradiction.Therefore, the theorem follows.

Theorem 4 . 4 .
For any integers n ≥ 6 and t with 2 ≤ t < n 2 , the only trees T n on n vertices of type DP2 with η p (T n ) = n − t are the ones with exactly one earring of size n − t − 1.Proof.This follows from Theorem 4.3 and the fact that β p (T n ) = n − t − 1, by using a partitionΠ := {{x 2 }, {x 3 }, • • • , {x n−t−1 }, {x, x 1 } ∪ B}, where {x 1 , • • • , x n−t−1 }are all the vertices of degree one adjacent to earring x and B is the set of all the remaining vertices in T n .

Theorem 4 . 5 .
If G is the graph in Figure6, then G is a graph of type DP2.