DDiscrete Mathematics and Theoretical Computer Science
DMTCS vol. (subm.) :1, by the authors, ( − k ) -critical trees and k -minimal trees Walid Marweni ∗ Faculty of Sciences - Sfax University, Tunisia.
In a graph G = ( V, E ) , a module is a vertex subset M of V such that every vertex outside M is adjacent to all ornone of M . For example, ∅ , { x } ( x ∈ V ) and V are modules of G , called trivial modules. A graph, all the modulesof which are trivial, is prime; otherwise, it is decomposable. A vertex x of a prime graph G is critical if G − x isdecomposable. Moreover, a prime graph with k non-critical vertices is called ( − k ) -critical graph. A prime graph G is k -minimal if there is some k -vertex set X of vertices such that there is no proper induced subgraph of G containing X is prime. From this perspective, I. Boudabbous proposes to find the ( − k ) -critical graphs and k -minimal graphs forsome integer k even in a particular case of graphs. This research paper attempts to answer I. Boudabbous’s question.First, it describes the ( − k ) -critical tree. As a corollary, we determine the number of nonisomorphic ( − k ) -critical treewith n vertices where k ∈ { , , (cid:98) n (cid:99)} . Second, it provide a complete characterization of the k -minimal tree. As acorollary, we determine the number of nonisomorphic k -minimal tree with n vertices where k ≤ . Keywords:
Graphs, tree, module, prime, critical vertex, minimal.
Our work falls within the framework of graph theory with a special focus on decomposition problems. A graph G = ( V ( G ) , E ( G )) (or G = ( V, E ) ) consists of a finite set V of vertices together with a set E ofpairs of distinct vertices, called edges . On the one hand, let G = ( V, E ) be a graph. The neighborhood of x in G , denoted by N G ( x ) or simply N ( x ) , is the set N G ( x ) = { y ∈ V \ { x } : { x, y } ∈ E } . The degree of x in G , denoted by d G ( x ) ( or d ( x ) ), is the cardinal of N G ( x ) . A vertex x with degree one is calleda leaf , its adjacent vertex is called a support vertex and it is denoted by x + . If x is a support vertex in G admitting a unique leaf neighbor, this leaf is denoted by x − . The set of leaves and support vertices in agraph G is denoted by L ( G ) and S ( G ) respectively. An internal vertex is a vertex with a degree greaterthan or equal to 2. The distance between two vertices u and v in G is the length (number of edge) of theshortest path connecting them and is denoted by dist G ( u, v ) or simply dist ( u, v ) . The notation x ∼ Y for each Y ⊆ V \{ x } means x is adjacent to all or none vertex of Y . The negation is denoted by x (cid:54)∼ Y .On the other hand, given a graph G = ( V, E ) , with each subset X of V , the graph G [ X ] = ( X, E ∩ ( X )) is an induced subgraph of G . For X ⊆ V (resp. x ∈ V ), the induced subgraph G [ V \ X ] (resp. G [ V \ { x } ] ) is denoted by G − X (resp. G − x ). The notions of isomorphism and embedding are definedin the following way. Two graphs G = ( V, E ) and G (cid:48) = ( V (cid:48) , E (cid:48) ) are isomorphic , which is denotedby G (cid:39) G (cid:48) , if there is an isomorphism from G onto G (cid:48) , i.e., a bijection from V onto V (cid:48) such that for ∗ [email protected] subm. to DMTCS © by the authors by the author(s) Distributed under a Creative Commons Attribution 4.0 International License a r X i v : . [ c s . D M ] F e b Walid Marweni all x, y ∈ V , { x, y } ∈ E if and only if { f ( x ) , f ( y ) } ∈ E (cid:48) . We say that a graph G (cid:48) embeds into agraph G if G (cid:48) is isomorphic to an induced subgraph of G . Otherwise, we say that G omits G (cid:48) . Givena graph G and a one-to-one function f defined on a set containing V ( G ) , we denote by f ( G ) the graph ( f ( V ( G )); f ( E ( G ))) = ( f ( V ( G )); {{ f ( x ) , f ( y ) } : { x, y } ∈ E ( G ) } ) . A nonempty subset C of V is a connected component of G if for x ∈ C and y ∈ V \ C, { x, y } / ∈ E and if for x (cid:54) = y ∈ C, there is asequence x = x , . . . , x n = y of C elements such that for each ≤ i ≤ n − , { x i , x i +1 } ∈ E . A vertex x of G is isolated if { x } constitutes a connected component of G . The set of the connected components of G is a partition of V , denoted C ( G ) . The graph G is connected if it has at most one connected componentof G . Otherwise, it is called non-connected . For example, a tree is a connected graph in which any twovertices are connected by exactly one path.In addition, let’s consider a graph G = ( V, E ) , a subset M of V is a module of G if every vertex out-side M is adjacent to all or none of M . This concept was introduced in Spinrad (1992) and independentlyunder the name interval in Cournier and Ille (1998); Fra¨ıss´e (1984); Schmerl and Trotter (1993) and an autonomous set in Ehrenfeucht and Rozenberg (1990). The empty set, the singleton sets, and the full set ofvertices are trivial modules . A graph is indecomposable (or primitive ) if all its modules are trivial; other-wise, it is decomposable. Therefore, indecomposable graphs with at least four vertices are prime graphs.This concept was developed in several papers e.g (Ehrenfeucht and Rozenberg (1990); Fra¨ıss´e (1984);Kelly (1986); Schmerl and Trotter (1993); Sumner (1973)), and is now elaborated in a book by Ehren-feucht, Harju and Rozenberg Ehrenfeucht et al. (1997). Properties of the prime substructures of a givenprime structure were determined by Schmerl and Trotter Schmerl and Trotter (1993) in their fundamentalpaper. Indeed, several papers within the same sphere of reference have then appeared (BELKHECHINEand BOUDABBOUS (2010); Belkhechine et al. (2010); Bouchaala et al. (2013); Boudabbous (2016);Boussa¨ıri and Cha¨ıchaˆa (2007); Ehrenfeucht et al. (1997); Ille (1997); Elayech et al. (2015); Pouzet andZaguia (2009); Sayar (2011)). For instance, the path defined on N n = { , ..., n } , denoted by P n , is primefor n ≥ . A path with extremities x and y is referred to as ( x, y ) -path. For example, it is easy to verifythat each prime graph is connected.The study of the hereditary aspect of the primality in the graphs revolve around the following generalquestion. Given a prime graph G , is there always a proper prime subgraph in G ? Addressing thisproblematic led to the publication of numerous papers. A first result in this direction dates back to D.P. Sumner Sumner (1973) who asserted that for every prime graph G , there exists X ⊆ V ( G ) such that | X | = 4 and G [ X ] is prime. In 1990, A. Ehrenfeucht and G. Rozenberg Ehrenfeucht and Rozenberg(1990) reported also that the prime graphs have the following ascendant hereditary property. Let X be asubset of a prime graph G such that G [ X ] is prime. If | V ( G ) \ X | ≥ , then there exist x (cid:54) = y ∈ V ( G ) \ X such that G [ X ∪ { x, y } ] is prime. The latter result was improved in 1993 by J. H. Schmerl and W. T.Trotter Schmerl and Trotter (1993) as follows: Each prime graph of order n , ( n ≥ ), embeds a primegraph of order n − . It is then natural to raise the next question. Given a prime graph G of order n ,is there always a prime subgraph of G of order n − ? The answer to this question is negative and theprime graph G such that G − x is decomposable for each x ∈ V ( G ) , referred to as critical graph , is thecounterexample. In 1993, J.H. Schmerl and W.T. Trotter Schmerl and Trotter (1993) characterized thecritical graphs.Consider now a prime graph G = ( V, E ) . A vertex x of G is said to be critical if G − x is decomposable.Otherwise, x is a non-critical vertex. The set of the non-critical vertices of G is denoted by σ ( G ) . More-over, if G admits k non-critical vertices, it is then called a ( − k ) -critical graph . Recently, Y.Boudabbousand Ille Boudabbous and Ille (2009) asked about the description of the ( − -critical graphs. Their ques- − k ) -critical trees and k -minimal trees minimal graphs defined as follows. A prime graph G is minimal for a vertex subset X , or X -minimal , if no proper induced subgraph of G containing X isprime. A graph G is k -minimal if it is minimal for some k -element set of k elements. A. Cournier andP. Ille Cournier and Ille (1998) in 1998 characterized the -minimal and -minimal graphs. Recently, M.Alzohairi and Y. Boudabbous characterized the 3-minimal triangle-free graphs Alzohairi and Boudabbous(2014). In 2015, M. Alzohairi characterized the triangle-free graphs which are minimal for some nonstable4-vertex subset Alzohairi (2015).Motivated by these two fundamental notions, I. Boudabbous proposes to find the ( − k ) -critical graphsand k -minimal graphs for some integer k even in a particular case of graphs. This work resolves what isrequested by I. Boudabbous. For this reason, we shall describe the prime tree having exactly k non-criticalvertices. Recall that (cid:98) x (cid:99) denotes the greatest integer ≤ x . Therefore, we obtain: Theorem 1.1
Let T = ( V, E ) be a tree with at least vertices and { x , ..., x k } be a vertex subset of G where k is an integer. T is ( − k ) -critical and σ ( T ) = { x , ..., x k } (see Figure 1 (a)), if and only if T satisfies the four assertions.1. For each x (cid:54) = y ∈ L ( T ) , dist ( x, y ) ≥ ,2. { x , ..., x k } ⊆ L ( T ) and ≤ k ≤ (cid:98) n (cid:99) ,3. For each x ∈ L ( T ) \{ x , ..., x k } , there is a unique i ∈ { , ..., k } such that dist ( x, x i ) = 3 and d ( x + ) = 2 ,4. If d ( x + i ) = 2 where i ∈ { , ..., k } , then for all x ∈ L ( T ) \{ x i } , dist ( x i , x ) ≥ . Moreover, we shall describe the k -minimal trees. As a matter of fact, we obtain: Theorem 1.2
Let T = ( V, E ) be a tree with at least vertices and let { x , ..., x k } be a vertex subset of G where k is a strictly positive integer. T is minimal for { x , ..., x k } (see Figure 1 (b)) if and only if T satisfies the three assertions.1. For each x (cid:54) = y ∈ L ( T ) , dist ( x, y ) ≥ .2. For each x ∈ L ( T ) , { x, x + } ∩ { x , ..., x k } (cid:54) = ∅ .3. If x i ∈ S ( T ) and x − i / ∈ { x , ..., x k } where i ∈ { , ..., k } , then d ( x i ) = 2 and there is j (cid:54) = i ∈{ , ..., k } such that x j ∈ L ( T ) and d ( x i , x j ) = 2 . We recall the characterization of the prime tree set forward by to M. Alzohairi and Y. Boudabbous.
Lemma 2.1 (Alzohairi and Boudabbous (2014))
1. If M is a nontrivial module in a decomposable tree T , then M is a stable set of T . Moreover, theelements of M are leaves of T . Walid Marweni
Fig. 1: ( a ) T is ( − -critical and σ ( T ) = { x , ..., x } . ( b ) T (cid:48) is minimal for { x , ..., x } .
2. A tree with at least four vertices is prime if and only if any two distinct leaves do not have the sameneighbor.
As an immediate consequence of Lemma 2.1, we have the following result.
Corollary 2.2
Let T = ( V, E ) be a tree. T is prime if and only if d ( x, y ) ≥ , for each x (cid:54) = y ∈ L ( T ) . The following observation follows immediately from Lemma 2.1.
Observation 2.3
Let T = ( V, E ) be a prime tree with n vertices. Then, |S ( T ) | = |L ( T ) | ≤ (cid:98) n (cid:99) . Now, we establish the next lemma that will be needed in the sequel.
Lemma 2.4
Let T = ( V, E ) be a prime tree and x ∈ L ( T ) . If T − x is decomposable, then there is y ∈ L ( T ) \{ x } such that { y, x + } is the unique module of T − x . Proof:
Consider a prime tree T = ( V, E ) and x ∈ L ( T ) . Assume that T − x is a decomposable tree.Resting upon Lemma 2.1, there exist two distinct leaves of T − x , said y and z , which have the sameneighbor. Hence, { y, z } is a module of T − x . Since T is a prime tree, x (cid:54)∼ { y, z } . Thus, x + ∈ { y, z } .Without loss of generality, we may assume that x + = z and we have y ∈ L ( T ) . Since T is prime and y ∈ L ( T ) , then dist ( y, u ) ≥ for each u (cid:54) = y ∈ L ( T ) . Therefore, { y, x + } is the unique module of T − x . (cid:50) Proof of Theorem 1.1.
Consider a tree T = ( V, E ) with n vertices where n ≥ and { x , ..., x k } is asubset of V where k is a strictly positive integer.Assume that T is ( − k ) -critical and σ ( T ) = { x , ..., x k } . Since T is prime, by Corollary 2.2, we have foreach x (cid:54) = y ∈ L ( T ) , dist ( x, y ) ≥ . Hence, T satisfies the condition (1) of Theorem 1.1.Moreover, let x ∈ V \L ( T ) , x is an internal vertex of T and T − x is a non-connected graph. Then, T − x is decomposable and x / ∈ σ ( T ) . Thus, { x , ..., x k } ⊆ L ( T ) . As T is prime, based on Observation 2.3,we have ≤ k ≤ (cid:98) n (cid:99) . Hence, T satisfies condition (2) of Theorem 1.1.Now, consider x ∈ L ( T ) \{ x , ..., x k } . Then, T − x is a decomposable tree. By Lemma 2.4, there is y ∈ L ( T ) \{ x } such that { y, x + } is the only module of T − x . Clearly, dist ( x, y ) = 3 and d ( x + ) = 2 .Now, prove that y ∈ σ ( T ) . To the contrary, suppose that y / ∈ σ ( T ) , implying that T − y is a decomposable − k ) -critical trees and k -minimal trees z ∈ L ( T ) \{ y } such that { z, y + } is the unique module of T − y .Thus, d ( y + ) = 2 and dist ( y, z ) = 3 . This implies that z = x and we obtain that T is with 4 vertices;which contradicts the fact that T is a tree having at least 5 vertices. Hence, y ∈ σ ( T ) . Therefore, T satisfies the condition (3) of Theorem 1.1.Besides, assume that there is i ∈ { , ..., k } such that d ( x + i ) = 2 . Then, T − x i is a prime tree and x + i ∈ L ( T − x i ) . Referring to Corollary 2.2, for all y ∈ L ( T − x i ) , dist ( x + i , y ) ≥ . Since L ( T − x i ) \{ x + i } = L ( T ) \{ x i } , then for each y (cid:54) = x i ∈ L ( T ) , dist ( x i , y ) ≥ . Hence, T satisfies condition (4)of Theorem 1.1.Conversely, assume that T satisfies the conditions (1)-(4) of Theorem 1.1. Proving that, T is ( − k ) -critical and σ ( T ) = { x , ..., x k } . Since for each x (cid:54) = y ∈ L ( T ) , dist ( x, y ) ≥ and by Corollary 2.2, T is prime. Clearly, if x ∈ V \L ( T ) , T − x is a non-connected graph. Thus, T − x is decomposable andhence x is a critical vertex.Furthermore, if x ∈ L ( T ) \{ x , ..., x k } , by assertion (3), there is a unique i ∈ { , ..., k } such that dist ( x, x i ) = 3 and d ( x + ) = 2 . Then, { x + , x i } is a module of T − x . Hence for each x ∈ L ( T ) \{ x , ..., x k } , T − x is decomposable and then x is a critical vertex.Now, given i ∈ { , ..., k } ; if d ( x + i ) ≥ , then L ( T ) \{ x i } = L ( T − x i ) . According to first hypothesis ofTheorem 1.1, for each x (cid:54) = y ∈ L ( T − x i ) , dist ( x, y ) ≥ . By Corollary 2.2, T − x i is a prime tree.Assume now that d ( x + i ) = 2 . Suppose that T − x i is a decomposable tree. Then, by Lemma 2.4 thereis a unique y (cid:54) = x i ∈ L ( T ) such that { y, x + i } is the unique module of T − x i . As a matter of fact, dist ( y, x i ) = 3 ; which contradicts the hypothesis 4 of Theorem 1.1. Hence, T − x i is prime. Conse-quently, T is ( − k ) -critical and σ ( T ) = { x , ..., x k } . (cid:50) Our second objective in this section lies in determining the number of nonisomorphic ( − k ) -criticaltrees with n ≥ vertices where k ∈ { , , (cid:98) n (cid:99)} . According to the characterization of critical graphsSchmerl and Trotter (1993), P is the a unique critical trees. To specify the the number of nonisomorphic ( − k ) -critical trees where k ∈ { , , (cid:98) n (cid:99)} , we introduce for all n ∈ N , the one-to-one function: T n : N → N p (cid:55)→ p + n Now, we introduce also the following trees. • For integers m ≥ , let A m +1 be the tree defined on { , ..., m } and E ( A m +1 ) = {{ , i } , { i, i + m } :1 ≤ i ≤ m } (see Figure 2). • For integers k ≥ , t ≥ , let P k,t be the tree defined on { , ..., t + k } and E ( P k,t ) = E ( T t ( P k )) ∪{{ i − , i } : 1 ≤ i ≤ t } ∪ {{ t + 2 , i } : 1 ≤ i ≤ t } (see Figure 3). • For integers m ≥ , n ≥ , n ≥ , for each p ∈ { , } , s p = n + ... + n p . Let P m,n ,n be thetree defined on { , ..., s + m } and E ( P m,n ,n ) = E ( T s ( P m )) ∪ {{ i − , i } : 1 ≤ i ≤ s } ∪ {{ s + 2 , i } , { s + m − , j } : 1 ≤ i ≤ n and n < j ≤ s } (see Figure 4). Proposition 2.5
1. Up to isomorphisms, the ( − -critical trees with n vertices are the tree P , n − where n is an even integer ≥ .2. Up to isomorphisms, the ( −(cid:98) n (cid:99) ) -critical trees with n vertices are the tree A n where n is an oddinteger ≥ . Walid Marweni
Fig. 2:
The tree A m +1 Fig. 3:
The tree P k,t Proof:
1. Clearly, departing from Theorem 1.1, P , n − is a ( − -critical tree where n ≥ and σ ( P , n − ) = { n − } . Now, we consider a ( − -critical tree T with n ≥ vertices such that σ ( T ) = { x } . ByTheorem 1.1, x ∈ L ( T ) . If x (cid:54) = x ∈ L ( T ) , then by assertion (3) of Theorem 1.1, dist ( x, x ) = 3 and d ( x + ) = 2 . On the contrary, suppose that |L ( T ) | = 2 . Since T is prime, then T is isomorphicto P ; which contradicts the fact that T has at least 5 vertices. Hence, |L ( T ) | ≥ . By assertion (3),for each y ∈ L ( T ) , dist ( y, x ) = 3 and d ( y + ) = 2 . Thus, T is isomorphic to P , n − where n ≥ is an even integer.2. By Theorem 1.1, A m +1 is ( −(cid:98) m + 12 (cid:99) ) -critical and σ ( A m +1 ) = L ( A m +1 ) where m ≥ .Now, we consider a ( −(cid:98) n (cid:99) ) -critical tree T with n ≥ vertices. Using Theorem 1.1, σ ( T ) ⊆ L ( T ) implies that | σ ( T ) | = (cid:98) n (cid:99) ≤ |L ( T ) | . By Observation 2.3, |L ( T ) | = |S ( T ) | ≤ (cid:98) n (cid:99) and therefore − k ) -critical trees and k -minimal trees Fig. 4:
The tree P m,n ,n |L ( T ) | = |S ( T ) | = (cid:98) n (cid:99) . Now, we shall prove that n is odd. To the contrary, suppose that n is even.Then, |L ( T ) | = |S ( T ) | = n . Hence, V ( T ) = L ( T ) ∪ S ( T ) . Since T [ S ( T )] is a tree, there exists avertex y ∈ S ( T ) with d T [ S ( T )] ( y ) = 1 . Hence, d T ( y ) = 2 . We may assume that N ( y ) = { y − , x } where x ∈ S ( T ) . Thus, { y, x − } is a module of T − y − ; which contradicts the fact that y − is not acritical vertex. Accordingly, n is odd, |L ( T ) | = |S ( T ) | = n − , and V ( T ) = L ( T ) ∪ S ( T ) ∪ { z } .We can assume that L ( T ) = { x , ..., x ( n − ) } and S ( T ) = { x +1 , ..., x +( n − ) } . Since T [ S ( T ) ∪ { z } ] is a tree, there exists a vertex x + i ∈ S ( T ) where ≤ i ≤ n − with d T [ S ( T )] ( x + i ) = 1 and hence d T ( x + i ) = 2 . By Theorem 1.1, dist ( x i , x j ) ≥ for each j (cid:54) = i ∈ { , ..., n − } . Hence, T isisomorphic to A n where n ≥ . (cid:50) As a consequence of Theorem 1.1, we get the following result.
Proposition 2.6
Up to isomorphisms, the ( − -critical trees with n ≥ vertices are the trees P n , P k,t where k ≥ , t ≥ and n = k + 2 t , and P m,n ,n where m ≥ , n , n ≥ and n = m + 2( n + n ) . Proof:
By Theorem 1.1, P n , P k,t where k ≥ , t ≥ , and P m,n ,n where m ≥ , n , n ≥ are ( − -critical trees and σ ( P n ) = { , n } , σ ( P k,t ) = { t + 1 , t + k } , and σ ( P m,n ,n ) = { s + 1 , s + m } .Now, assume that T is a ( − -critical tree with n ≥ vertices such that σ ( T ) = { x , x } . By Theorem1.1, x , x ∈ L ( T ) . As T is a prime tree, then the ( x , x ) -path is isomorphic to P k where k ≥ . If |L ( T ) | = 2 , then T is isomorphic to P n and n = k . Assume that |L ( T ) | ≥ , then by Theorem 1.1, foreach x ∈ L ( T ) \{ x , x } ; there is a unique i ∈ { , } such that dist ( x, x i ) = 3 and d ( x + ) = 2 . Hence, T is isomorphic to P k,t where k ≥ , t ≥ and n = k + 2 t or T is isomorphic to P k,n ,n where k ≥ , n , n ≥ and n = k + 2( n + n ) . (cid:50) Walid Marweni
Theorem 2.7
The number of nonisomorphic ( − -critical trees with n vertices equals: • (cid:106) n (cid:107) − if n ≡ mod . • (cid:106) n (cid:107) if n ≡ mod . • (cid:106) n (cid:107) (cid:16)(cid:106) n (cid:107) + 1 (cid:17) − if n ≡ mod . • (cid:106) n (cid:107) (cid:16)(cid:106) n (cid:107) + 1 (cid:17) otherwise. Proof:
At the beginning, it is not difficult to verify that there are not two isomorphic different trees in theunion { P m : m ≥ } ∪ { P k,t : k ≥ , t ≥ } ∪ { P m,n ,n : m ≥ , n ≥ and n ≥ } .By Proposition 2.6, P is the unique ( − -critical tree with five vertex and P is the unique ( − -criticaltree with six vertices. As a matter of fact, the result holds.Now, assume that n ≥ . By Proposition 2, the nonisomorphic ( − -critical trees with n vertices are P n ,the family of P k,t where t ≥ , k ≥ , and n = 2 t + k , or the family of P m,n ,n where ≤ n ≤ n , m ≥ , and n = 2( n + n ) + m . Therefore, it is sufficient to determine the number of the family of P k,t and the number of the family of P m,n ,n .Let S m = { ( n , n ) ∈ N × N : 1 ≤ n ≤ n , n + n = n − m } , where ≤ m ≤ n − and let C t = { k ∈ N : 5 ≤ k and k = n − t } , where ≤ t ≤ n − . Since n − m = 2( n + n ) , it is obviousthat n and m are of the same parity. Hence, we distinguish two cases. Case 1: If n = 2 p where ≤ p and m = 2 q where ≤ q ≤ p − .Consider S = p − (cid:91) q =2 S q and C = p − (cid:91) t =1 C t . First, it is clear that the number of the family of P k,t is thecardinality of the set C . Moreover, it is clear that | C t | = 1 where ≤ t ≤ p − . Hence, | C | = p − (cid:88) t =1 | C t | = p − . Second, obviously the number of the family of P m,n ,n is the cardinality of the set S . Furthermore,we have | S | = p − (cid:88) q =2 | S q | . It is noticeable that for each ≤ q ≤ p − , | S q | = P ( n − q ) , where P i ( j ) isthe number of partitions of j to i parts. Recall that for an integer k ≥ , P ( k ) = (cid:98) k (cid:99) where (cid:98) x (cid:99) is the − k ) -critical trees and k -minimal trees ≤ x Anderson (2006). We get then | S | = p − (cid:88) q =2 P (cid:18) n − q (cid:19) = p − (cid:88) q =2 (cid:22) n − q (cid:23) = p − (cid:88) q =2 (cid:22) n − q (cid:23) = p − (cid:88) i =0 (cid:22) i (cid:23) . = ( k − if p = 2 k, ( k − k if p = 2 k + 1 . Case 2:
Let n = 2 p + 1 where ≤ p and m = 2 q + 1 where ≤ q ≤ p − . Let S = p − (cid:91) q =2 S q +1 and C = p − (cid:91) t =1 C t . Clearly, the number of the family of P k,t is the cardinality of the set C . Hence, | C | = p − (cid:88) t =1 | C t | = p − . In addition, the number of the family of P m,n ,n is the cardinality of the set S .Therefore, we have | S | = p − (cid:88) q =2 | S q +1 | .Since for each ≤ q ≤ p − , | S q +1 | = P (cid:16) ( n − − q (cid:17) . Proceeding in the same manner as case 1, if p = 2 k where k ≥ , then | S | = ( k − . Otherwise, | S | = ( k − k .Consequently, the number of nonisomorphic ( − -critical trees with n vertices equals: (cid:106) n (cid:107) − if n ≡ mod , (cid:106) n (cid:107) if n ≡ mod , (cid:106) n (cid:107) (cid:16)(cid:106) n (cid:107) + 1 (cid:17) − if n ≡ mod , (cid:106) n (cid:107) (cid:16)(cid:106) n (cid:107) + 1 (cid:17) if n ≡ mod . (cid:50) Walid Marweni
We set two major objectives throughout this section. First, to characterize the k -minimal trees. Second,to determine the number of nonisomorphic k -minimal trees with n vertices where k ∈ { , , } . Proof of Theorem 1.2.
Let T = ( V, E ) be a tree with n ≥ vertices and let { x , ..., x k } be a vertexsubset of G where k is a strictly positive integer. Assume that T is minimal for { x , ..., x k } . Since T isprime, it satisfies the first condition of Theorem 1.2. Suppose, on the contrary, that there is y ∈ L ( T ) suchthat { y, y + } ∩ { x , ..., x k } = ∅ . Since T − y is a decomposable tree, by Lemma 2.4, there is x ∈ L ( T ) such that { x, y + } is a module of T − y . Thus, d ( y + ) = 2 and d ( x, y ) = 3 . If x / ∈ { x , ..., x k } , then T − x is a decomposable tree. By using again Lemma 2.4, d ( x + ) = 2 and so T is isomorphic to P ;which contradicts the fact that n ≥ .Moreover, assume that x ∈ { x , ..., x k } and d ( x + ) ≥ , then L ( T ) \{ y } = L ( T − { y, y + } ) . By Lemma2.1, T − { y, y + } is a prime tree containing { x , ..., x k } ; which contradicts the fact that T is minimal for { x , ..., x k } . Hence, for each x ∈ L ( T ) , { x, x + } ∩ { x , ..., x k } (cid:54) = ∅ and T satisfies the second conditionof Theorem 1.2.Now, assume that there is x i ∈ S ( T ) and x − i / ∈ { x , ..., x k } where ≤ i ≤ k . On the contrary,suppose that d ( x i ) ≥ , then L ( T ) \{ x − i } = L ( T − x − i ) . By Lemma 2.1, T − x − i is a prime treecontaining { x , ..., x k } ; which is impossible. Hence, d ( x i ) = 2 . On the contrary, suppose that for each j (cid:54) = i ∈ { , ..., k } such that x j ∈ L ( T ) , d ( x i , x j ) ≥ . Since T − x − i is a decomposable tree, thenby Lemma 2.4 there is y ∈ L ( T ) such that d ( x i , y ) = 2 and hence y / ∈ { x , ..., x k } . By Condition 2, y + ∈ { x , ..., x k } and so d ( y + ) = 2 . Thus, T is isomorphic to P ; which is impossible. Therefore, T satisfies the third condition.Conversely, let T = ( V, E ) be a tree with n ≥ vertices. Since for each x (cid:54) = y ∈ L ( T ) , d ( x, y ) ≥ , T is a prime tree. Let X be a subset of V such that { x , ..., x k } ⊆ X and T [ X ] is prime. Consider x ∈ L ( T ) . If x ∈ { x , ..., x k } , then x ∈ X .Now, assume that x / ∈ { x , ..., x k } . On the contrary, suppose that x / ∈ X . By assertion (2) of Theorem1.2, x + ∈ { x , ..., x k } . Since T satisfies assertion (3) of Theorem 1.2, then d ( x + ) = 2 and there is i ∈ { , ..., k } such that x i ∈ L ( T ) and d ( x i , x + ) = 2 . Thus, { x + , x i } is a module of T [ X ] ; whichis impossible. Hence, x ∈ X . We conclude that L ( T ) ⊂ X . Since T [ X ] is a prime, it is connected.Therefore, T [ X ] is a tree containing L ( T ) . Hence, X = V . Thus, T is minimal for { x , ..., x k } . (cid:50) The following corollary is an immediate consequence of Theorem 1.2.
Corollary 3.1
For any distinct vertices x , x ,..., and x k in a prime tree H , there is an induced subtree T of H that contains { x , x , ..., x k } , and satisfies the assertions of Theorem 1.2. Our second objective is to determine the number of nonisomorphic k -minimal trees with n verticeswhere k ∈ { , , } . According to the characterization of 1-minimal and 2-minimal graphs, P is theunique 1-minimal tree and P k , where k ≥ , is the unique 2-minimal tree Cournier and Ille (1998).To specify the number of nonisomorphic -minimal trees with n vertices, we introduce the followingtree. • For positive integers k , m , n with k ≤ m ≤ n , let S k,m,n be the ( k + m + n + 1) -vertex tree with theunion of the paths of lengths k , m , and n having common endpoint r . Let a , ..., a k , b , ..., b m , and c , ..., c n denote the other vertices on these paths, indexed by their distance from r (see Figure 5).As an immediate consequence of Theorem 1.2, we get the following result which was already obtainedby M. Alzohairi and Y. Boudabbous in Alzohairi and Boudabbous (2014). − k ) -critical trees and k -minimal trees Fig. 5: S k,m,n Corollary 3.2
Let x , y , and z be distinct vertices in a tree T . The tree T is minimal for { x, y, z } if andonly if T and { x, y, z } have one of the following forms:1. T (cid:39) P .2. T (cid:39) P k with k ≥ such that { x, y, z } contains the leaves.3. T (cid:39) S k,m,n with m ≥ such that x , y , and z are the leaves.4. T (cid:39) S , ,n such that { x, y, z } = { a , b , c n } .5. T (cid:39) S , , such that { x, y, z } = { a , b , c } . Proposition 3.3
The number of nonisomorphic -minimal trees with n vertices equals: • n ∈ { , } . • n = 6 . • (cid:104) ( n − (cid:105) − (cid:4) n − (cid:5) + (cid:4) n − (cid:5) − if n ≥ , where [ x ] is the nearest integer from x . Proof:
At the beginning, it is not difficult to verify that there are not two isomorphic different graphs inthe union { P k : k ≥ } ∪ { S k,m,n : m = 2 } .By Corollary 3.2, P is the unique 3-minimal tree with four vertices and P is the unique 3-minimal treewith five vertices. In addition, the only -minimal tree with six vertices are P and S , , . Therefore, theresult holds for n ∈ { , , } .Now, assume that n ≥ . By Corollary 3.2, the non isomorphic -minimal n -vertex trees are P n and thefamily of S k,m,t , where k ≤ m ≤ t , m ≥ , and k + m + t + 1 = n . Therefore, it is sufficient to provethat the cardinality of the set S = { ( k, m, t ) ∈ N × N × N : 1 ≤ k ≤ m ≤ t, m ≥ , k + m + t = n − } equals (cid:20) ( n − (cid:21) − (cid:22) n − (cid:23) + (cid:22) n − (cid:23) − . Walid Marweni
Let S = { ( k, m, t ) ∈ S : k ≥ } . It is easy to infer that | S − S | = P ( n − − . Notice that | S | = |{ ( p, q, r ) ∈ N × N × N : 1 ≤ p ≤ q ≤ r, p + q + r = n − }| = P ( n − . Moreover, groundedon Anderson (2006), the number of partitions of k with at most parts is equal to (cid:104) ( k +3) (cid:105) . It followsthat P ( k ) = (cid:104) ( k +3) (cid:105) − (cid:4) k (cid:5) − Anderson (2006). Therefore, | S | = (cid:20) ( n − (cid:21) − (cid:22) n − (cid:23) − and | S − S | = (cid:22) n − (cid:23) − . Thus, | S | = (cid:20) ( n − (cid:21) − (cid:22) n − (cid:23) + (cid:22) n − (cid:23) − . (cid:50) The problems of finding the ( − k ) -critical graphs and the k -minimal graphs seem to be challenging where k is an integer ( k ≥ ). At least, we solve these problems in the particular case of trees. In addition, wedetermine the number of nonisomorphic ( − k ) -critical trees with n ≥ vertices where k ∈ { , , (cid:98) n (cid:99)} .Besides, we specify the number of nonisomorphic -minimal trees with n ( n ≥ ) vertices. References
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