On the Distinguishing number of Functigraphs
aa r X i v : . [ m a t h . C O ] D ec ON THE DISTINGUISHING NUMBER OF FUNCTIGRAPHS
MUHAMMAD FAZIL, MUHAMMAD MURTAZA, USMAN ALI, IMRAN JAVAID
Abstract.
Let G and G be disjoint copies of a graph G , and let g : V ( G ) → V ( G ) be a function. A functigraph F G consists of the vertex set V ( G ) ∪ V ( G )and the edge set E ( G ) ∪ E ( G ) ∪ { uv : g ( u ) = v } . In this paper, we extendthe study of the distinguishing number of a graph to its functigraph. We discussthe behavior of the distinguishing number in passing from G to F G and find itssharp lower and upper bounds. We also discuss the distinguishing number offunctigraphs of complete graphs and join graphs. Preliminaries
Given a key ring of apparently identical keys to open different doors, how manycolors are needed to identify them? This puzzle was given by Rubin [23] in 1980 forthe first time. In this puzzle, there is no need for coloring to be proper. Indeed, onecannot find a reason why adjacent keys must be assigned different colors, whereasin other problems like storing chemicals, scheduling meetings a proper coloring isneeded, and one with a small number of colors is required.From the inspiration of this puzzle, Albertson and Collins [1] introduced theconcept of the distinguishing number of a graph as follows: A labeling f : V ( G ) →{ , , , ..., 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 , denotedby Dist ( G ), is the least integer t such that G has t -distinguishing labeling. Forexample, the distinguishing number of a complete graph K n is n , the distinguishingnumber of a path P n is 2 and the distinguishing number of a cycle C n , n ≥ 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 someof the vertices with one or more colors and same for the edges, labeling vertices oredges, adding or deleting vertices or edges) of destroying the symmetries of a graph.Collins and Trenk defined the distinguishing chromatic number in [13] where theyused proper t -distinguishing for vertex labeling. They have also given a comparisonbetween the distinguishing number, the distinguishing chromatic number and the Key words and phrases. distinguishing number, functigraph.2010
Mathematics Subject Classification. ∗ Corresponding author: [email protected]. hromatic number for families like complete graphs, paths, cycles, Petersen graphand trees etc. Kalinowski and Pilsniak [20] have defined similar graph parameters,the distinguishing index and the distinguishing chromatic index, they labeled edgesinstead of vertices. They have also given a comparison between the distinguishingnumber 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 provedthat a graph is t -distinguishable if and only if it has a determining set that is ( t − K n : k with n ≥ k ≥ G considered in this paper are simple,non-trivial and connected. The open neighborhood of a vertex u of G is N ( u ) = { v ∈ V ( G ) : uv ∈ E ( G ) } and the closed neighborhood of u is N ( u ) ∪ { u } . Two vertices u, v are adjacent twins if N [ u ] = N [ v ] 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 ofvertices of G . The induced subgraph , denoted by < S > , is the graph whose vertexset is S and whose edge set is the set of all those edges in E ( G ) which have bothend vertices in S .The idea of permutation graph was introduced by Chartrand and Harary [10] forthe first time. They defined the permutation graph as follows: a permutation graphconsists of two identical disjoint copies of a graph G , say G and G , along with | V ( G ) | additional edges joining V ( G ) and V ( G ) according to a given permutationon { , , ..., | V ( G ) |} . Dorfler [14], introduced a mapping graph which consists of twodisjoint identical copies of graph where the edges between the two vertex sets arespecified by a function. The mapping graph was rediscovered and studied by Chenet al. [12], where it was called the functigraph. A functigraph is an extension ofpermutation graph. Formally the functigraph is defined as follows: Let G and G be disjoint copies of a connected graph G , and let g : V ( G ) → V ( G ) be a function.A functigraph F G of a graph G consists of the vertex set V ( G ) ∪ V ( G ) and the edgeset E ( G ) ∪ E ( G ) ∪ { uv : g ( u ) = v } . Linda et al. [15, 16] and Kang et al. [21] havestudied the functigraph for some graph invariants like metric dimension, dominationand 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 ) = A , V ( G ) = B , g : A → B is afunction, g ( V ( G )) = I , | g ( V ( G )) | = | I | = s .This paper is organized as follows. In Section 2, we give sharp lower and upperbounds for distinguishing number of functigraph. This section also establishes the onnections between the distinguishing number of graphs and their correspondingfunctigraphs in the form of realizable results. In Section 3, we provide the distin-guishing number of functigraphs of complete graphs and join of path graphs. Someuseful results related to these families have also been presented in this section.2. Bounds and some realizable results
The sharp lower and upper bounds on the distinguishing number of functigraphsare given in the following result.
Proposition 2.1.
Let G be a connected graph of order n ≥ , then ≤ Dist ( F G ) ≤ Dist ( G ) + 1 . Both bounds are sharp.Proof.
Obviously, 1 ≤ Dist ( F G ) by definition. Let Dist ( G ) = t and f be a t -distinguishing labeling for graph G . Also, let u i ∈ A and v i ∈ B , 1 ≤ i ≤ n . Weextend labeling f to F G as: f ( u i ) = f ( v i ) for all 1 ≤ i ≤ n . We have following twocases for g :(1) If g is not bijective, then f as defined earlier is a t -distinguishing labeling for F G . Hence, Dist ( F G ) ≤ t .(2) If g is bijective, then f as defined earlier destroys all non-trivial automor-phisms of F G except the flipping of G and G in F G , for some choices of g .Thus, Dist ( F G ) ≤ t + 1.For the sharpness of the lower bound, take G = P and g : A → B , be a functionsuch that g ( u i ) = v , i = 1 , g ( u ) = v . For the sharpness of the upper bound,take G as rigid graph and g as identity function. (cid:3) Since at least m colors are required to break all automorphisms of a twin set ofcardinality m , so we have the following corollary. Proposition 2.2.
Let U , U , ..., U t be disjoint twin sets in a connected graph G oforder n ≥ and m = max {| U i | : 1 ≤ i ≤ t } ,(i) Dist ( G ) ≥ m .(ii) If Dist ( G ) = m , then Dist ( F G ) ≤ m . Lemma 2.3.
Let G be a connected graph of order n ≥ and g be a constant function,then Dist ( F G ) = Dist ( G ) .Proof. Let I = { v } ⊂ B . Then Γ( G ) = Γ( < A ∪ { v } > ) ⊂ Γ( F G ). Thus, vertices in A ∪ { v } are labeled by Dist ( G ) colors. Since g is a constant function, therefore allvertices in V ( F G ) \ { A ∪ { v }} are not similar to any vertex in A ∪ { v } in functigraph F G . Therefore, vertices in V ( F G ) \ { A ∪ { v }} can also be labeled from these Dist ( G )colors. Hence, Dist ( F G ) = Dist ( G ). (cid:3) emark 2.4. Let G be a connected graph and Dist ( F G ) = m if g is constant and Dist ( F G ) = m if g is not constant, then m ≥ m . 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 calledthe twin stems .We define a function ψ : N \{ } → N \{ } as ψ ( m ) = k where k is the least numbersuch that m ≤ (cid:0) k (cid:1) + k . For example, ψ (19) = 5. Note that ψ is well-defined. Lemma 2.5.
If a graph G has t ≥ twin stems of length 2 rooted at same majorvertex, then Dist ( G ) ≥ ψ ( t ) .Proof. Let x ∈ V ( G ) be a major vertex and xu i u ′ i where 1 ≤ i ≤ t are twin stems oflength 2 attach with x . Let H = < { x, u i , u ′ i } > and k = ψ ( t ). We define a labeling f : V ( H ) → { , , ..., k } as: f ( x ) = k, (1) f ( u i ) = if ≤ i ≤ k if k + 1 ≤ i ≤ k if k + 1 ≤ i ≤ k ... ... k if ( k − k + 1 ≤ i ≤ k (2) f ( u ′ i ) = (cid:26) i mod( k ) if ≤ i mod( k ) ≤ k − ,k if i mod( k ) = 0 , Using this labeling, one can see that f is a t -distinguishing for H . Since permu-tations with repetition of k colors, when 2 of them are taken at a time is equal to2 (cid:0) k (cid:1) + 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 ). (cid:3) Lemma 2.6.
For any integer t ≥ , 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 : x x x ...x ( t − +1 be a path.Join ( t − + 1 twin stems x u i u ′ i where 1 ≤ i ≤ ( t − + 1 each of lengthtwo with vertex x of P ( t − +1 . This completes construction of G . We first showthat Dist ( G ) = t . For t = 2, we have two twin stems attach with x , and hence Dist ( G ) = 2. For t ≥
3, we define a labeling f : V ( G ) → { , , , ..., t } as follows: '1u1u'2u2u'3 u3x1x2 y1y2 v'1v'2v'3v' v1v2v3vu'(t-1)2+1u(t-1)2+1 (t-1)2+1(t-1)2 + Figure 1.
Graph with
Dist ( G ) = t = Dist ( F G ) .f ( x i ) = t , for all i , where 1 ≤ i ≤ ( t − + 1 .f ( u i ) = if ≤ i ≤ t − , if t ≤ i ≤ t − , if t − ≤ i ≤ t − , ... ... t − if ( t − t −
2) + 1 ≤ i ≤ ( t − ,t if i = ( t − + 1 .f ( u ′ i ) = i mod( t − if ≤ i mod( t − ≤ t − and i = ( t − + 1 ,t − if i mod( t −
1) = 0 ,t if i = ( t − + 1 . Using this labeling, one can see the unique automorphism preserving this labelingis the identity automorphism. Hence, f is a t -distinguishing. Since permutationwith repetition of t − (cid:0) t − (cid:1) + ( t − t − +1 twin stems can be labeled by at least t -colors. Hence, t is the leastinteger such that G has t -distinguishing labeling. Now, we denote the correspondingvertices of G as v i , v ′ i , y i for all i , where 1 ≤ i ≤ ( t − + 1 and construct afunctigraph F G by defining g : V ( G ) → V ( G ) as follows: g ( u i ) = g ( u ′ i ) = y i , forall i , where 1 ≤ i ≤ ( t − + 1 and g ( x i ) = g ( y i ), for all i , where 1 ≤ i ≤ ( t − + 1as shown in the Figure 1. Thus, F G has only symmetries of ( t − + 1 twin stemsattach with y . Hence, Dist ( F G ) = t. (cid:3) Consider an integer t ≥
4. We construct graph G similarly as in proof of Lemma2.6 by taking a path P ( t − +1 : x x ...x ( t − +1 and attach ( t − + 1 twin stems x u i u ′ i where 1 ≤ i ≤ ( t − + 1 with any one of its end vertex say x . Usingsimilar labeling and arguments as in proof of Lemma 2.6 one can see that f is t − istinguishing and t − G has t − F G , where g : V ( G ) → V ( G ) is defined by: g ( u i ) = g ( u ′ i ) = y i ,for all i , where 1 ≤ i ≤ ( t − + 1, g ( x i ) = v i , for all i , where 1 ≤ i ≤ ( t − − g ( x i ) = y i , for all i , where ( t − ≤ i ≤ ( t − + 1. From this construction, F G has only symmetries of 2 twin stems attach with y , and hence Dist ( F G ) = 2. Thus,we have the following result which shows that Dist ( G ) + Dist ( F G ) can be arbitrarylarge: Lemma 2.7.
For any integer t ≥ , 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 bytaking a path P t − +1 : x x ...x t − +1 and attach 4( t − + 1 twin stems x u i u ′ i ,where 1 ≤ i ≤ t − + 1 with x . Using similar labeling and arguments as inproof of Lemma 2.6 one can see that f is 2 t − t − G has 2 t − g as g ( u i ) = g ( u ′ i ) = y i , for all i , where 1 ≤ i ≤ t − + 1, g ( x i ) = v i , for all i , where1 ≤ i ≤ t − t and g ( x i ) = y i , for all i , where 3 t − t + 1 ≤ i ≤ t − + 1 . Thus, F G has only symmetries of ( t − + 1 twin stems attach with y , and hence Dist ( F G ) = t −
1. After making this type of construction, we have the followingresult which shows that
Dist ( G ) − Dist ( F G ) can be arbitrary large: Lemma 2.8.
For any integer t ≥ , 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 ofgraphs
In this section, we discuss the distinguishing number of functigraphs on completegraphs, edge deletion graphs of complete graph and join of path graphs.Let G be the complete graph of order n ≥ A and B be its two copies. Weuse following terminology for F G in proof of Theorem 3.3: Let I = { v , v , ..., 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 , ≤ i ≤ s }| . From the definitions of l and m , we notethat 2 ≤ l ≤ n − s + 1 and 0 ≤ m ≤ s − ψ ( m ) as defined in previous section, we have following lemma: Lemma 3.1.
Let G be the complete graph of order n ≥ and g be a bijectivefunction, then Dist ( F G ) = ψ ( n ) .Proof. Let A = { u , u , ..., u n } and I = { g ( u ) , g ( u ) , ..., g ( u n ) } = B . Also let k = ψ ( n ). Let f : V ( F G ) → { , , ..., k } be a labeling in which f ( u i ) is defined asin equation (1) and f ( g ( u i )) as in equation (2) in proof of Lemma 2.5. Using thislabeling one can see that f is a k -distinguishing labeling for F G . Since permutation ith repetition of k colors, when 2 of them are taken at a time is equal to 2 (cid:0) k (cid:1) + k ,therefore at least k colors are needed to label the vertices in F G . Hence, k is theleast integer for which F G has k -distinguishing labeling. (cid:3) 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 , u , ..., u m ∈ A are those vertices of A such that g ( u i ) = g ( u j ) where 1 ≤ i = j ≤ m in B . Also ( u i u j )( g ( u i ) g ( u j )) ∈ Γ( F G ) for all i = j where 1 ≤ i, j ≤ m . 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 havefollowing proposition: Proposition 3.2.
Let G be a complete graph of order n ≥ and g be a functionsuch that ≤ m ≤ s , then Dist ( F G ) ≥ ψ ( m ) . The following result gives the distinguishing number of functigraphs of completegraphs.
Theorem 3.3.
Let G = K n be the complete graph of order n ≥ , and let < s ≤ n − , then Dist ( F G ) ∈ { n − s, n − s + 1 , ψ ( m ) } . Proof.
We discuss following cases for l :(1) If l = n − s + 1 >
2, then A contains n − s + 1 twin vertices and B contains n − s twin vertices (except for n = 3 , B contains no twin vertices).Also, there are m (= s −
1) vertices in A which have distinct images in B .These m vertices and their distinct images are labeled by at least ψ ( m ) colors(only 1 color if m = 1) by Proposition 3.2. Since n − s + 1 is the largestamong n − s + 1, n − s and ψ ( m ). Thus, n − s + 1 is the least number suchthat F G has ( n − s + 1)- distinguishing labeling. Thus, Dist ( F G ) = n − s + 1.(2) If l = n − s + 1 = 2, then ψ ( m ) ≥ max { n − s + 1 , n − s } , and hence Dist ( F G ) = ψ ( m ).(3) If l < n − s , then B contains largest set of n − s twin vertices in F G . Also,there are m ( ≤ s −
2) vertices in A each of which has distinct image in B .Since n − s ≥ ψ ( m ), therefore Dist ( F G ) = n − s .(4) If l = n − s >
2, then both A and B contain largest set of n − s twin verticesin F G . Also, there are m (= s −
2) vertices in A which have distinct imagesin B . Since n − s ≥ ψ ( m ), therefore Dist ( F G ) = n − s .(5) If l = n − s = 2, then we take two subcases:(a) If 1 < s ≤ ⌊ n ⌋ + 1, then both A and B contain largest set of n − s twinvertices in F G . Also, there are m (= s −
2) vertices in A which havedistinct images in B . Since n − s ≥ ψ ( m ) (if ψ ( m ) exists), therefore Dist ( F G ) = n − s .(b) If ⌊ n ⌋ + 1 < s ≤ n −
1, then ψ ( m ) ≥ max { n − s + 1 , n − s } , and hence Dist ( F G ) = ψ ( m ). Let e ∗ be an edge of a connected graph G . Let G − ie ∗ is the graph obtained bydeleting i edges from graph G. A vertex v of a graph G is called saturated if it isadjacent to all other vertices of G .We define a function φ : N → N \ { } as φ ( i ) = k , where k is the least numbersuch that i ≤ (cid:0) k (cid:1) . For instance, φ (32) = 9. Note that φ is well defined. Theorem 3.4.
Let G be the complete graph of order n ≥ and G i = G − ie ∗ forall i where ≤ i ≤ ⌊ n ⌋ and e ∗ joins two saturated vertices of the graph G . If g is aconstant function, then Dist ( F G i ) = max { n − i, φ ( i ) } . Proof.
On deleting i edges e ∗ from G , we have n − i saturated vertices and i twinsets each of cardinality two. We will now show that exactly φ ( i ) colors are requiredto label vertices of all i twin sets. We observe that, a vertex in a twin set can bemapped on any one vertex in any other twin set. Since two vertices in a twin setare labeled by a unique pair of colors out of (cid:0) k (cid:1) pairs of k colors, therefore at least k colors are required to label vertices of i twin sets. Now, we discuss the followingtwo cases for φ ( i ):(1) If φ ( i ) ≤ n − i , then number of colors required to label n − i saturatedvertices is greater than or equal to number of colors required to label verticesof i twin sets. Thus, we label n − i saturated vertices with exactly n − i colors and out of these n − i colors, φ ( i ) colors will be used to label verticesof i twin sets.(2) If φ ( i ) > n − i , then number of colors required to label n − i saturatedvertices is less than the number of colors required to label vertices of i twinsets. Thus, we label vertices of i twin sets with φ ( i ) colors and out of these φ ( i ) colors, n − i 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, Dist ( F G i ) = Dist ( G i ) . (cid:3) Suppose that G = ( V , E ) and G ∗ = ( V , E ) be two graphs with disjoint vertexsets V and V and disjoint edge sets E and E . The join of G and G ∗ is the graph G + G ∗ , in which V ( G + G ∗ ) = V ∪ V and E ( G + G ∗ ) = E ∪ E ∪ { uv : u ∈ V , v ∈ V } . Theorem 3.5. [5]
Let G and G ∗ be two connected graphs, then Dist ( G + G ∗ ) ≥ max { Dist ( G ) , Dist ( G ∗ ) } . Proposition 3.6.
Let P n be a path graph of order n ≥ , then for all m, n ≥ and < s < m + n , ≤ Dist ( F P m + P n ) ≤ .Proof. Let P m : v , ..., v m and P n : u , ..., u n . We discuss following cases for m, n .
1) If m = 2 and n = 2, then P + P = K , and hence 1 ≤ Dist ( F K ) ≤ m = 2 and n = 3, then P + P has 3 saturated vertices. Thus, 1 ≤ Dist ( F P + P ) ≤ s where 2 ≤ s ≤ g in F P + P , one can see 1 ≤ Dist ( F P + P ) ≤ m = 3 and n = 3, then a labeling f : V ( P + P ) → { , , } defined as: f ( x ) = if x = v , v if x = v , u if x = u , u is a distinguishing labeling for P + P , and hence Dist ( P + P ) = 3. Thus,1 ≤ Dist ( F P + P ) ≤ s where 2 ≤ s ≤ g in F P + P , one can see 1 ≤ Dist ( F P + P ) ≤ m ≥ n ≥
4, then a labeling f : V ( P m + P n ) → { , } defined as: f ( x ) = (cid:26) if x = v , u , ..., u n if x = u , v , ..., 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. (cid:3) References [1] M. O. Albertson and K. L. Collins, Symmetry breaking in graphs,
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