On the Distinguishing number of Functigraphs

Let $G_{1}$ and $G_{2}$ be disjoint copies of a graph $G$, and let $g:V(G_{1})\rightarrow V(G_{2})$ be a function. A functigraph $F_{G}$ consists of the vertex set $V(G_{1})\cup V(G_{2})$ and the edge set $E(G_{1})\cup E(G_{2})\cup \{uv:g(u)=v\}$. In this paper, we extend the study of the distinguishing number of a graph to its functigraph. We discuss the behavior of the distinguishing number in passing from $G$ to $F_{G}$ and find its sharp lower and upper bounds. We also discuss the distinguishing number of functigraphs of complete graphs and join graphs.


Preliminaries
Given a key ring of apparently identical keys to open different doors, how many colors are needed to identify them? This puzzle was given by Rubin [23] in 1980 for the first time. In this puzzle, there is no need for coloring to be proper. Indeed, one cannot find a reason why adjacent keys must be assigned different colors, whereas in other problems like storing chemicals, scheduling meetings a proper coloring is needed, and one with a small number of colors is required.
From the inspiration of this puzzle, Albertson and Collins [1] introduced the concept of the distinguishing number of a graph as follows: A labeling f : V (G) → {1, 2, 3, ..., t} is called a t-distinguishing if no non-trivial automorphism of a graph G preserves the vertex labels. The distinguishing number of a graph G, denoted by Dist(G), is the least integer t such that G has t-distinguishing labeling. For example, the distinguishing number of a complete graph K n is n, the distinguishing number of a path P n is 2 and the distinguishing number of a cycle C n , n ≥ 6 is 2. For a graph G of order n, 1 ≤ Dist(G) ≤ n [1]. If H is a subgraph of a graph G such that automorphism group of H is a subset of automorphism group of G, then Dist(H) ≤ Dist(G).
Harary [18] gave different methods (orienting some of the edges, coloring some of the vertices with one or more colors and same for the edges, labeling vertices or edges, adding or deleting vertices or edges) of destroying the symmetries of a graph. Collins and Trenk defined the distinguishing chromatic number in [13] where they used proper t-distinguishing for vertex labeling. They have also given a comparison between the distinguishing number, the distinguishing chromatic number and the chromatic number for families like complete graphs, paths, cycles, Petersen graph and trees etc. Kalinowski and Pilsniak [20] have defined similar graph parameters, the distinguishing index and the distinguishing chromatic index, they labeled edges instead of vertices. They have also given a comparison between the distinguishing number and the distinguishing index for a connected graph G of order n ≥ 3. Boutin [7] introduced the concept of determining sets. In [4], Albertson and Boutin proved that a graph is t-distinguishable if and only if it has a determining set that is (t − 1)distinguishable. They also proved that every Kneser graph K n:k with n ≥ 6 and k ≥ 2 is 2-distinguishable. A considerable literature has been developed in this area see [2,3,6,8,9,11,19,22,24].
Unless otherwise specified, all the graphs G considered in this paper are simple, non-trivial and connected. The open neighborhood of a vertex u of G is and non adjacent twins if N(u) = N(v). If u, v are adjacent or non adjacent twins, then u, v are twins. A set of vertices is called twin-set if every of its two vertices are twins. A graph H is said to be a subgraph of a graph G if V (H) ⊆ V (G) and E(H) ⊆ E(G). Let S ⊂ V (G) be any subset of vertices of G. The induced subgraph, denoted by < S >, is the graph whose vertex set is S and whose edge set is the set of all those edges in E(G) which have both end vertices in S.
The idea of permutation graph was introduced by Chartrand and Harary [10] for the first time. They defined the permutation graph as follows: a permutation graph consists of two identical disjoint copies of a graph G, say G 1 and G 2 , along with |V (G)| additional edges joining V (G 1 ) and V (G 2 ) according to a given permutation on {1, 2, ..., |V (G)|}. Dorfler [14], introduced a mapping graph which consists of two disjoint identical copies of graph where the edges between the two vertex sets are specified by a function. The mapping graph was rediscovered and studied by Chen et al. [12], where it was called the functigraph. A functigraph is an extension of permutation graph. Formally the functigraph is defined as follows: Let G 1 and G 2 be disjoint copies of a connected graph G, and let g : A functigraph F G of a graph G consists of the vertex set V (G 1 )∪V (G 2 ) and the edge set E(G 1 ) ∪ E(G 2 ) ∪ {uv : g(u) = v}. Linda et al. [15,16] and Kang et al. [21] have studied the functigraph for some graph invariants like metric dimension, domination and zero forcing number. In [17], we have studied the fixing number of functigraph. The aim of this paper is to study the distinguishing number of functigraph.
Throughout the paper, we will denote the set of all automorphisms of a graph G by Γ(G), the functigraph of G by F G , V (G 1 ) = A, V (G 2 ) = B, g : A → B is a function, g(V (G 1 )) = I, |g(V (G 1 ))| = |I| = s. This paper is organized as follows. In Section 2, we give sharp lower and upper bounds for distinguishing number of functigraph. This section also establishes the connections between the distinguishing number of graphs and their corresponding functigraphs in the form of realizable results. In Section 3, we provide the distinguishing number of functigraphs of complete graphs and join of path graphs. Some useful results related to these families have also been presented in this section.

Bounds and some realizable results
The sharp lower and upper bounds on the distinguishing number of functigraphs are given in the following result.
Both bounds are sharp.
We have following two cases for g: (1) If g is not bijective, then f as defined earlier is a t-distinguishing labeling for (2) If g is bijective, then f as defined earlier destroys all non-trivial automorphisms of F G except the flipping of G 1 and G 2 in F G , for some choices of g. Thus, Dist(F G ) ≤ t + 1. For the sharpness of the lower bound, take G = P 3 and g : A → B, be a function such that g(u i ) = v 1 , i = 1, 2 and g(u 3 ) = v 3 . For the sharpness of the upper bound, take G as rigid graph and g as identity function.
Since at least m colors are required to break all automorphisms of a twin set of cardinality m, so we have the following corollary.  A vertex v of degree at least three in a connected graph G is called a major vertex. Two paths rooted from the same major vertex and having the same length are called the twin stems.
We define a function ψ : Lemma 2.5. If a graph G has t ≥ 2 twin stems of length 2 rooted at same major vertex, then Dist(G) ≥ ψ(t).
We define a labeling f : V (H) → {1, 2, ..., k} as: Using this labeling, one can see that f is a t-distinguishing for H. Since permutations with repetition of k colors, when 2 of them are taken at a time is equal to 2 k 2 +k, therefore at least k colors are needed to label the vertices in t-stems. Hence, k is the least integer for which G has k-distinguishing labeling. Since Γ(H) ⊆ Γ(G), therefore Dist(G) ≥ ψ(t).
Lemma 2.6. For any integer t ≥ 2, there exists a connected graph G and a function g such that Dist(G) = t = Dist(F G ).
Proof. Construct the graph G as follows: let P (t−1) 2 +1 : x 1 x 2 x 3 ...x (t−1) 2 +1 be a path. Join (t − 1) 2 + 1 twin stems x 1 u i u ′ i where 1 ≤ i ≤ (t − 1) 2 + 1 each of length two with vertex x 1 of P (t−1) 2 +1 . This completes construction of G. We first show that Dist(G) = t. For t = 2, we have two twin stems attach with x 1 , and hence Dist(G) = 2. For t ≥ 3, we define a labeling f : V (G) → {1, 2, 3, ..., t} as follows: x 3 x (t-1) 2 +1 y 3 Using this labeling, one can see the unique automorphism preserving this labeling is the identity automorphism. Hence, f is a t-distinguishing. Since permutation with repetition of t − 1 colors, when 2 of them are taken at a time is 2 t−1 2 + (t − 1), therefore (t−1) 2 +1 twin stems can be labeled by at least t-colors. Hence, t is the least integer such that G has t-distinguishing labeling. Now, we denote the corresponding vertices of G 2 as v i , v ′ i , y i for all i, where 1 ≤ i ≤ (t − 1) 2 + 1 and construct a functigraph F G by defining g : V (G 1 ) → V (G 2 ) as follows: g(u i ) = g(u ′ i ) = y i , for all i, where 1 ≤ i ≤ (t − 1) 2 + 1 and g(x i ) = g(y i ), for all i, where 1 ≤ i ≤ (t − 1) 2 + 1 as shown in the Figure 1. Thus, F G has only symmetries of (t − 1) 2 + 1 twin stems attach with y 1 . Hence, Dist(F G ) = t.
Consider an integer t ≥ 4. We construct graph G similarly as in proof of Lemma 2.6 by taking a path P (t−3) 2 +1 : x 1 x 2 ...x (t−3) 2 +1 and attach (t − 3) 2 + 1 twin stems x 1 u i u ′ i where 1 ≤ i ≤ (t − 3) 2 + 1 with any one of its end vertex say x 1 . Using similar labeling and arguments as in proof of Lemma 2.6 one can see that f is t − 2 distinguishing and t − 2 is least integer such that G has t − 2 distinguishing labeling. Define functigraph F G , where g : From this construction, F G has only symmetries of 2 twin stems attach with y 1 , and hence Dist(F G ) = 2. Thus, we have the following result which shows that Dist(G) + Dist(F G ) can be arbitrary large: Lemma 2.7. For any integer t ≥ 4, there exists a connected graph G and a function g such that Dist(G) + Dist(F G ) = t.
Consider t ≥ 3. We construct graph G similarly as in proof of Lemma 2.6 by taking a path P 4(t−1) 2 +1 : x 1 x 2 ...x 4(t−1) 2 +1 and attach 4(t − 1) 2 + 1 twin stems Using similar labeling and arguments as in proof of Lemma 2.6 one can see that f is 2t − 1 distinguishing and 2t − 1 is the least integer such that G has 2t − 1 distinguishing labeling. Let us now define g as Thus, F G has only symmetries of (t − 2) 2 + 1 twin stems attach with y 1 , and hence Dist(F G ) = t − 1. After making this type of construction, we have the following result which shows that Dist(G) − Dist(F G ) can be arbitrary large: Lemma 2.8. For any integer t ≥ 3, there exists a connected graph G and a function g such that Dist(G) − Dist(F G ) = t.

The distinguishing number of functigraphs of some families of graphs
In this section, we discuss the distinguishing number of functigraphs on complete graphs, edge deletion graphs of complete graph and join of path graphs.
Let G be the complete graph of order n ≥ 3 and A and B be its two copies. We use following terminology for F G in proof of Theorem 3.3: Let I = {v 1 , v 2 , ..., v s } and n i = |{u ∈ A : g(u) = v i }| for all i, where 1 ≤ i ≤ s. Also, let l = max{n i : 1 ≤ i ≤ s} and m = |{n i : n i = 1, 1 ≤ i ≤ s}|. From the definitions of l and m, we note that 2 ≤ l ≤ n − s + 1 and 0 ≤ m ≤ s − 1.
Using function ψ(m) as defined in previous section, we have following lemma: Lemma 3.1. Let G be the complete graph of order n ≥ 3 and g be a bijective function, then Dist(F G ) = ψ(n).
Proof. Let A = {u 1 , u 2 , ..., u n } and I = {g(u 1 ), g(u 2 ), ..., g(u n )} = B. Also let k = ψ(n). Let f : V (F G ) → {1, 2, ..., k} be a labeling in which f (u i ) is defined as in equation (1) and f (g(u i )) as in equation (2) in proof of Lemma 2.5. Using this labeling one can see that f is a k-distinguishing labeling for F G . Since permutation with repetition of k colors, when 2 of them are taken at a time is equal to 2 k 2 + k, therefore at least k colors are needed to label the vertices in F G . Hence, k is the least integer for which F G has k-distinguishing labeling.
Let G be a complete graph and let g : A → B be a function such that 2 ≤ m ≤ s. Without loss of generality assume u 1 , u 2 , ..., u m ∈ A are those vertices of A such that By using similar labeling f as defined in Lemma 3.1, at least ψ(m) color are needed to break these automorphism in F G . Thus, we have following proposition: Proposition 3.2. Let G be a complete graph of order n ≥ 3 and g be a function such that 2 ≤ m ≤ s, then Dist(F G ) ≥ ψ(m).
The following result gives the distinguishing number of functigraphs of complete graphs.
Theorem 3.3. Let G = K n be the complete graph of order n ≥ 3, and let 1 < s ≤ n − 1, then Proof. We discuss following cases for l: Let e * be an edge of a connected graph G. Let G − ie * is the graph obtained by deleting i edges from graph G. A vertex v of a graph G is called saturated if it is adjacent to all other vertices of G.
We define a function φ : N → N \ {1} as φ(i) = k, where k is the least number such that i ≤ k 2 . For instance, φ(32) = 9. Note that φ is well defined. Theorem 3.4. Let G be the complete graph of order n ≥ 5 and G i = G − ie * for all i where 1 ≤ i ≤ ⌊ n 2 ⌋ and e * joins two saturated vertices of the graph G. If g is a constant function, then Proof. On deleting i edges e * from G, we have n − 2i saturated vertices and i twin sets each of cardinality two. We will now show that exactly φ(i) colors are required to label vertices of all i twin sets. We observe that, a vertex in a twin set can be mapped on any one vertex in any other twin set. Since two vertices in a twin set are labeled by a unique pair of colors out of k 2 pairs of k colors, therefore at least k colors are required to label vertices of i twin sets. Now, we discuss the following two cases for φ(i): (1) If φ(i) ≤ n − 2i, then number of colors required to label n − 2i saturated vertices is greater than or equal to number of colors required to label vertices of i twin sets. Thus, we label n − 2i saturated vertices with exactly n − 2i colors and out of these n − 2i colors, φ(i) colors will be used to label vertices of i twin sets. (2) If φ(i) > n − 2i, then number of colors required to label n − 2i saturated vertices is less than the number of colors required to label vertices of i twin sets. Thus, we label vertices of i twin sets with φ(i) colors and out of these φ(i) colors, n − 2i colors will be used to label saturated vertices in G i . If g is constant, then by using same arguments as in the proof of Lemma 2.3, Suppose that G = (V 1 , E 1 ) and G * = (V 2 , E 2 ) be two graphs with disjoint vertex sets V 1 and V 2 and disjoint edge sets E 1 and E 2 . The join of G and G * is the graph G + G * , in which V (G + G * ) = V 1 ∪ V 2 and E(G + G * ) = E 1 ∪ E 2 ∪ { uv: u ∈ V 1 , v ∈ V 2 }.
Proof. Let P m : v 1 , ..., v m and P n : u 1 , ..., u n . We discuss following cases for m, n.
(4) If m ≥ 2 and n ≥ 4, then a labeling f : V (P m + P n ) → {1, 2} defined as: .., v m is a distinguishing labeling for P m + P n , and hence Dist(P m + P n ) = 2. Thus, result follows by Proposition 2.1.