Vertex-addition strategy for domination-like invariants

In [J. Graph Theory 13 (1989) 749–762], McCuaig and Shepherd gave an upper bound of the domination number for connected graphs with minimum degree at least two. In this paper, we propose a simple strategy which, together with the McCuaig-Shepherd theorem, gives a sharp upper bound of the domination number via the number of leaves. We also apply the same strategy to other domination-like invariants, and find a relationship between such invariants and the number of leaves.


Introduction 1.Domination concept and our strategy
All graphs considered in this paper are finite, simple, and undirected.Let G be a graph.For u ∈ V (G), we let d G (u), N G (u) and N G [u] denote the degree, the open neighborhood and the closed neighborhood of u, respectively; thus d G (u) = |N G (u)| and N G [u] = N G (u) ∪ {u}.We let δ(G) and ∆(G) denote the minimum degree and the maximum degree of G, respectively.A vertex u ∈ V (G) is a leaf of G if the degree of u in G is exactly one.We let L(G) denote the set of leaves of G.An edge of G is called a pendant edge if the edge is incident with a leaf of G.For two subsets X, Y of V (G), we say that X dominates Y if Y ⊆ u∈X N G [u].A subset of V (G) which dominates V (G) is called a dominating set of G.The minimum cardinality of a dominating set of G is called the domination number of G, and is denoted by γ(G).
The domination number is one of the important invariants in Graph Theory, and it can be widely applied to real problems, for example, school bus routing problem, social network theory and the location of radio stations (see [10,11]).To meet various additional requirements for above problems, many domination-like concepts were defined and studied.
We first introduce an orthodox flow of research of domination-like concepts by citing the original result on domination.The following is a well-known result given by Ore.
Theorem A (Ore [17]) Let G be a connected graph of order n ( 2).Then γ(G) The upper bound of γ(G) in Theorem A is best possible.Furthermore, the connected graphs G attaining the equality in Theorem A were characterized by Fink, Jacobson, Kinch and Roberts [7] and Payan and Xuong [18] as follows (here the corona of a graph H is the graph obtained from H by adding a pendant edge to each vertex of H).
Theorem B (Fink et al. [7]; Payan and Xuong [18]) Let G be a connected graph of order n.Then γ(G) = n 2 if and only if G is either C 4 or the corona of a connected graph.
In particular, any connected graphs G with γ(G) = |V (G)| 2 except for C 4 have some leaves.Thus one may suspect that the domination number of many connected graphs G with δ(G) 2 is much less than the half of |V (G)|.For connected graphs with minimum degree at least two, McCuaig and Shepherd [16] showed the following theorem (here B is the set consisting of graphs depicted in Figure 1).
B 6 Figure 1: The graphs belonging to B Theorem C (McCuaig and Shepherd [16]) Let G be a connected graph of order n with δ(G) 2. Then either G ∈ B or γ(G) the electronic journal of combinatorics 24(3) (2017), #P3.45 Considering Theorem C, we know that the existence of leaves is a cause of an increase of the domination number.On the other hand, it seems that Theorem C gives no insight to general graphs G (which do not require the condition δ(G) 2).Thus one problem naturally arises: Find a relationship between the domination number and the number of leaves.As a related result to the problem, for example, Favaron [6] gave the following theorem.
Theorem D (Favaron [6]) Let l 2 be an integer, and let T be a tree of order n having exactly l leaves.Then γ(T ) n+l 3 .However, there exist infinitely many connected (non-tree) graphs G with n vertices, l leaves and γ(G) > n+l 3 (see Theorem 2.1 in Section 2).Thus, when we study a relationship between domination and leaves in general graphs, it is insufficient to only consider trees.
In order to get a desired relation, we propose a simple vertex-addition strategy as follows: For a given connected graph G, (S1) we construct a new graph H with δ(H) 2 which is obtained from G by adding a special graph to each leaf of G, (S2) give a small dominating set S of H which is assured by Theorem C, and (S3) reduce S to a dominating set of G.
In Section 2, we show the following theorem by the above vertex-addition strategy.Note that Theorem 1.1 is a common generalization of Theorems A and C. In Section 2, we also give a generalization of Theorem B by using the proof technique of Theorem C.
We return to general domination-like concepts.For many domination-like invariants µ, the same upper bounds were found: (D1) A sharp upper bound of µ(G) for every connected graph G (of large order).
(D2) A sharp upper bound of µ(G) for every connected graph G with δ(G) 2 (with finite exceptions).
In many cases, the following results are also given.
(D1 ′ ) A characterization of connected graphs G attaining the equality of the bound in (D1).
Our main aim in this paper is to give the following new steps for an invariant µ by using the vertex-addition strategy.
(D3) A sharp upper bound of µ(G) for every connected graph G with l leaves (i.e., a common generalization of (D1) and (D2)).
In general, the results such as (D2) and (D2 ′ ) tend to be independently shown from (D1) and (D1 ′ ).Thus our strategy might give an alternative proof of the results such as (D1) and (D1 ′ ).
As we mentioned above, domination-like invariants were widely defined.Thus it is difficult to deal with all of them.In this paper, as typical domination-like invariants, we deal with two especially famous invariants, namely, total domination and Roman domination in Sections 3 and 4, respectively.Our main results for such invariants are Theorems 3.2 and 4.2.

Definitions
Our notation and terminology are standard, and mostly taken from [5].Exceptions are as follows.
For n 3, we let P n and C n denote the path and the cycle of order n, respectively.A vertex u of a connected graph H is a central vertex if for any v ∈ V (H), the distance from u to v is at most the radius of H.Note that a path of odd order has exactly one central vertex, and a path of even order has exactly two central vertices.
A graph G is l-leaf minimally connected if (L2) G has exactly l leaves, and (L3) for each e ∈ E(G), either e is a bridge of G or G − e has at least l + 1 leaves.
Note that a graph G is 0-leaf minimally connected if and only if G is a connected graph with δ(G) 2 and δ(G − e) = 1 for any non-bridge edge e ∈ E(G).The following fact clearly holds.Fact 1.1 Let l 0 be an integer, and let G be a connected graph of order n ( 2) having exactly l leaves.Then G has a spanning l-leaf minimally connected subgraph.
The following lemma will be used in the proof of our main results.
the electronic journal of combinatorics 24(3) (2017), #P3.45 Lemma 1.2 Let G be an l-leaf minimally connected graph, and let H be a graph obtained from G by adding for each v ∈ L(G), a new graph C v and some edges between N G [v] and V (C v ).Let e ∈ E(G) be an edge incident with no leaves of G. Then e is a bridge of H or |L(H)| < |L(H − e)|.
Proof.We may assume that e is not a bridge of H.By the construction of H, e is not a bridge of G. Since G is l-leaf minimally connected, it follows that G − e has at least l + 1 leaves.In particular, e is incident with a vertex x with d G (x) = 2. Write e = xx ′ and N G (x) = {x ′ , y}.Suppose that d H (x) 3. Since d G (x) = 2 and e is incident with no leaves of G, this implies that y is a leaf of G, and hence e is a bridge of G, which is a contradiction.Thus d H (x) = 2, and so x is a leaf of H − e. Consequently we have Let C be a connected graph, and let v ∈ V (C).The (C, v)-corona of a graph G is the graph obtained from G by adding a copy C u of C to each vertex u ∈ V (G) and identifying u and the vertex of C u corresponding with v (see Figure 2).Note that if is a path of order 2, the (C, v)-corona of a graph G is exactly the corona of G (defined in the paragraph preceding Theorem B in Subsection 1.1).
Let A = {A 1 , . . ., A m } be a set of vertex-disjoint graphs with some attachment vertices.(For example, we will use a set of some copies of U i in the first paragraph of Section 2.) A graph G is minimally connected with respect to A if (M1) G is obtained from 1 i m A i by adding m − 1 edges e 1 , . . ., e m−1 , (M2) each edge e j joins an attachment vertex of A i and an attachment vertex of A i ′ for some i, i ′ (i = i ′ ), and Note that if A is a set of copies of a graph A with exactly one attachment vertex v, then a minimally connected graph with respect to A is the (A, v)-corona of a tree.

Domination
Let U 0 , U 1 and U 2 be the graphs depicted in Figure 3.We define the attachment vertices of U i as the vertices of U i enclosed with a circle.Let l 0 be an integer.A set A of vertex-disjoint graphs is (γ, l)-optimal if Let F l be the set of minimal-connected graphs with respect to a (γ, l)-optimal set.Then we can easily check that every graph in F l is l-leaf minimally connected.Let R be the set of graphs depicted in Figure 4.In the proof of Theorem C, McCuaig and Shepherd [16] showed the following theorem.[16]) Let G be a 0-leaf minimally connected graph of order n.Then γ(G)

Now we give a natural generalization of Theorem E (by using Theorem E).
Theorem 2.1 Let l 0 be an integer, and let G be an l-leaf minimally connected graph of order n ( 3).Then γ(G) if and only if either l = 0 and the electronic journal of combinatorics 24(3) (2017), #P3.45 Proof.If l = 0, then by Theorem E, the desired result holds.Thus we may assume that l 1.If G ∈ F l , then we can easily check that γ(G) = 2n+l 5 .Thus it suffices to show that if γ(G) and edges vx v , vz v (see Figure 5).Then δ(H) 2 and H has a bridge 2 and the construction of H, we see that H is 0-leaf minimally connected.This together with Theorem E leads to γ(H) . Let S be a dominating set of H with |S| = γ(H), and let Since γ(G) 2n+l 5 , the equality of (2.1) holds.In particular, γ(H) = 2|V (H)|

5
. Since This completes the proof of Theorem 2.1.
Proof of Theorem 1.1.Let G be a connected graph of order n ( 3) having exactly l leaves.Since every leaf of G is adjacent to a vertex in the electronic journal of combinatorics 24(3) (2017), #P3.45 By Fact 1.1, G has a spanning l-leaf minimally connected subgraph H. Since deleting an edge cannot decrease the domination number, we have γ(G) γ(H).This together with Theorem 2.1 implies that either l = 0 and H ∈ {B 1 , B 2 , B 3 } or γ(G) γ(H) , and so H is the corona of a tree.By tedious arguments, this implies that G is either C 4 or the corona of a connected graph.Thus we also get Theorem B as a corollary of Theorem 2.1.Now we argue a sharpness of Theorem 1.1.Fix an integer l 0. Let n be an integer with n max{2l, 3} and n + 3l ≡ 0 (mod 5), and let A be a (γ, l)-optimal set with |A| = n+3l 5 .Then every minimal-connected graph G with respect to A has n vertices and l leaves, and satisfies γ(G) = 2n+l 5 .Thus Theorem 1.1 for the case where 0 l n 2 is best possible.
Let n be an integer with max{l + 1, 3} n 2l − 1.Let L 1 be a star having exactly 2l − n + 1 leaves, and for each 2 i n − l, let L i be a star of order 2. For each i (1 i n − l), we define the attachment vertex of L i as one of the central vertices of L i .Then every minimal-connected graph G with respect to {L i : 1 i n − l} has n vertices and l leaves, and satisfies γ(G) = n − l.Thus Theorem 1.1 for the case where l n − 1 is best possible.

Total domination
In this section, we find a relationship between total domination and the number of leaves.

Definition and known results
Let G be a graph without isolated vertices.For two subsets X, Y of V (G), we say that X totally dominates The minimum cardinality of a total dominating set of G is called the total domination number of G, and is denoted by γ t (G).The concept of total domination was introduced in [3], and has been actively studied (see a book [13]).
A study of upper bounds of the total domination number derives from the following theorem proved by Cockayne, Dawes and Hedetniemi [3].
Theorem F (Cockayne et al. [3]) Let G be a connected graph of order n ( 3).Then γ t (G) Theorem G (Brigham et al. [1]) Let G be a connected graph of order n.Then γ t (G) = 2n 3 if and only if G is either C 3 or C 6 or the (P 3 , v)-corona of a connected graph where v is an endvertex of P 3 .
For graphs with minimum degree at least two, Henning [12] gave a sharp upper bound of the total domination number as follows (here B t is the set consisting of graphs depicted in Figure 6).
The graphs belonging to B t Theorem H (Henning [12]) Let G be a connected graph of order n with δ(G) Indeed, he showed a stronger theorem than Theorem H.In order to state his result, we give a further definition.
Let U 0 t and U 1 t be the graphs depicted in Figure 7.We define the attachment vertex of U i t as the vertex of U i t enclosed with a circle.
The graphs belonging to R t Let F l t be the set of minimal-connected graphs with respect to a (γ t , l)-optimal set.Then we can easily check that every graph in F l t is l-leaf minimally connected.Let R t be the set of graphs depicted in Figure 8.
Henning [12] showed the following theorem, which gives Theorem H as a corollary.
Theorem I (Henning [12]) Let G be a 0-leaf minimally connected graph of order n.

Main result for total domination
The main result in this section is the following.
Theorem 3.1 Let l 0 be an integer, and let G be an l-leaf minimally connected graph of order n ( 3).Then γ t (G) if and only if either l = 0 and .Thus it suffices to show that if γ t (G) 4n+2l 7 , then G ∈ F l t .For v ∈ L(G), let u v be the unique neighbor of v. Let H be the graph obtained from G by adding for each v ∈ L(G), a new path Q v = x v y v z v w v and edges vx v , u v w v (see Figure 9).Then δ(H) 2 and H  . Let S be a total dominating set of H with |S| = γ t (H).Then the following claim holds.
) cannot totally dominate {u v } in H.This leads to the desired conclusion.
Thus we have We show that S 0 is a total dominating set of G. Since S is a total dominating set of H and Furthermore, by the definition of T v , the electronic journal of combinatorics 24(3) (2017), #P3.45 It follows from (3.3) and (3.4) that (3.2) holds.Since u ∈ {u v : v ∈ L(G)} is arbitrary, we have This together with the fact that Theorem 3.2 Let l 0 be an integer, and let G be a connected graph of order n ( 3) having exactly l leaves.Then either l = 0 and G ∈ B t or the electronic journal of combinatorics 24(3) (2017), #P3.45 Proof.If l = n − 1, then G is a star, and hence γ t (G) = 2, as desired.Thus we may assume that l n − 2. Then |V (G) − L(G)| 2 and each vertex in V (G) is adjacent to a vertex in V (G) − L(G).In particular, V (G) − L(G) is a total dominating set of G, and so By Fact 1.1, G has a spanning l-leaf minimally connected subgraph H. Since deleting an edge cannot decrease the total domination number, we have γ t (G) γ t (H).This together with Theorem 3.1 implies that either l = 0 and H ∈ {B  t , and so H is the (P 3 , v)-corona of a tree where v is an endvertex of P 3 .By tedious arguments, this implies that G is either C 3 or C 6 or the (P 3 , v)-corona of a connected graph.Thus we also get Theorem G as a corollary of Theorem 3.1.Now we argue a sharpness of Theorem 3.2.Fix an integer l 0. Let n be an integer with n max{3l, 4} and n + 4l ≡ 0 (mod 7), and let A be a (γ t , l)-optimal set with |A| = n+4l 7 .Then every minimal-connected graph G with respect to A has n vertices and l leaves, and satisfies γ t (G) = 4n+2l 7 .Thus Theorem 3.2 for the case where 0 l n 3 is best possible.Assume that l 2, and let n be an integer with 2l n 3l − 1.For each i (1 i n − 2l), let L i be a path of order 3, and for each i (n − 2l + 1 i l), let L i be a path of order 2. For each i (1 i l), we define the attachment vertex of L i as one of the endvertices of L i .Then every minimal-connected graph G with respect to {L i : 1 i l} has n vertices and l leaves, and satisfies γ t (G) = n − l.Thus Theorem 3.2 for the case where n+1 3 l n 2 is best possible.Let n be an integer with l + 2 n 2l − 1.Let L ′ 1 be a star having exactly 2l − n + 1 leaves, and for each 2 i n − l, let L ′ i be a star of order 2. For each i (1 i n − l), we define the attachment vertex of L ′ i as one of the central vertices of L ′ i .Then every minimal-connected graph G with respect to {L ′ i : 1 i n − l} has n vertices and l leaves, and satisfies γ t (G) = n − l.Thus Theorem 3.2 for the case where l n − 2 is best possible.Moreover, since the total domination number of a star is 2, Theorem 3.2 for the case where l = n − 1 is best possible.

Roman domination
In this section, we find a relationship between Roman domination and the number of leaves.

Definition and known results
Let G be a graph.A function f : The minimum weight of a Roman dominating function of G is called the Roman domination number of G, and is denoted by γ R (G).The Roman domination number was introduced by Stewart [19], and was studied by Cockayne, Dreyer Jr., Hedetniemi and Hedetniemi, [4] in earnest.Recently, various properties on the Roman domination number has been explored in, for example, [8,9,14,15].
Chambers, Kinnersley, Prince and West [2] gave a sharp upper bound of the Roman domination number for connected graphs with a characterization of the graphs attaining the equality.
Theorem J (Chambers et al. [2]) Let G be a connected graph of order n ( 3).Then γ R (G) 4n 5 .Theorem K (Chambers et al. [2]) Let G be a connected graph of order n.Then γ R (G) = 4n 5 if and only if G is either C 5 or the (P 5 , v)-corona of a connected graph where v is the unique central vertex of P 5 .
They also proved the following theorem (here B R is the set consisting of graphs depicted in Figure 10).
R be the graphs depicted in Figure 11.We define the attachment vertex of U i R as the vertex of U i R enclosed with a circle.
the electronic journal of combinatorics 24(3) (2017), #P3.45 R be the set of minimal-connected graphs with respect to a (γ R , l)-optimal set.Then we can easily check that every graph in F l R is l-leaf minimally connected.Chambers et al. [2] proved the following theorem, which gives Theorem L as a corollary.
Theorem M (Chambers et al. [2]) Let G be a 0-leaf minimally connected graph of order n.Then γ R (G)

Main result for Roman domination
The main result in this section is the Theorem 4.1 Let l 0 be an integer, and let G be an l-leaf minimally connected graph of order n ( 3).Then γ R (G) if and only if either l = 0 and Proof.If l = 0, then by Theorem M, the desired result holds.Thus we may assume that 12).Then δ(H) 2 and Furthermore, by Lemma 1.2 and the construction of H, we see that H is 0-leaf minimally connected.This together with Theorem M leads to γ R (H) Then the following claim holds.
Lemma 4.3 Let l 2 be an integer, and let G be a connected graph having exactly l leaves.Then there exists a subgraph H of G such that each component of H is a spider and L(H) = L(G).
Proof.For u, v ∈ L(G) with u = v, a path P of G joining u and v is a spider with L(P ) ⊆ L(G).Thus there exists a subgraph H of G such that each component of H is a spider and L(H) ⊆ L(G).Choose H so that |L(H)| is as large as possible.Suppose that L(H) = L(G), and let Let P y be the path of P z joining w and y.Then T 2 = P y ∪ Q is a path of G with endvertices w and x.In particular, T 2 is a spider with L(T Proof of Theorem 4.2.Let G be a connected graph of order n ( 3) having exactly l leaves.It is known that γ R (G 1 ) 2γ(G 1 ) for any graph G 1 (see [4]).This together with (2.2) in the proof of Theorem R , and so H is the (P 5 , v)-corona of a tree where v is the unique central vertex of P 5 .By tedious arguments, this implies that G is either C 5 or the (P 5 , v)-corona of a connected graph.Thus we also get Theorem K as a corollary of Theorem 4.1.
the electronic journal of combinatorics 24(3) (2017), #P3.45 Now we argue a sharpness of Theorem 4.2.Fix an integer l 0. Let n be an integer with n max{ 5l 2 , 5} and n + 3l ≡ 0 (mod 11), and let A be a (γ R , l)-optimal set with |A| = n+3l 11 .Then every minimal-connected graph G with respect to A has n vertices and l leaves, and satisfies γ R (G) = 8n+2l 11 .Thus Theorem 4.2 for the case where 0 l 2n 5 is best possible.Assume that l is even, and let n be an integer with 2l n 5l 2 − 1.For each i (1 i n − 2l), let L i be a path of order 5, and for each i (n − 2l + 1 i l 2 ), let L i be a path of order 4. For each i (1 i

Theorem 1 . 1
Let l 0 be an integer, and let G be a connected graph of order n ( 3) having exactly l leaves.Then either l = 0 and G ∈ B or

Figure 4 :
The graphs belonging to R

2n+l 5 . 5 ,Remark 1
Since B is the set of graphs B containing one of B 1 , B 2 and B 3 as a spanning subgraph and satisfying γ(B) > 2|V (B)| this together with (2.2) implies that either l = 0 and H ∈ B or γ(G) min{ 2n+l 5 , n − l}.Consequently Theorem 1.1 holds.In the proof of Theorem 1.1, we further assume that γ(G) n 2 .Then it follows from Theorem 2.1 that l = n 2 and H ∈ F n 2

4n+2l 7 . 1 .
Proof.If l = 0, then by Theorem I, the desired result holds.Thus we may assume that l If G ∈ F l t , then we can easily check that γ t (G) = 4n+2l 7

Figure 9 :
Figure 9: Construction of H has a cutvertex u v where v is a leaf of G.In particular, H ∈ {B1 t , B 2 t , B 3 t , B 4 t } ∪ R t .Furthermore, by Lemma 1.2 and the construction of H, we see that H is 0-leaf minimally connected.This together with Theorem I leads to γ t (H)

2n 3 .
Then it follows from Theorem 3.1 that l = n 3 and H ∈ F n 3

Figure 10 :
Figure 10: The graphs belonging to B R

1 ) 4 . 2
a) (Otherwise).It follows from Claim 4.1 and the definition of f 0 , we havef 0 (a) f (a) for all a ∈ V (G) − L(G).(4.Claim The function f 0 is a Roman dominating function of G.Proof.Let p ∈ V (G) be a vertex with f 0 (p) = 0.It suffices to show that p is adjacent to a vertex of G assigned 2 by f 0 .If p ∈ L(G), then we have f 0 (u p ) = g(u p ) = 2; if p ∈ V (G) − {v, u v : v ∈ L(G)}, then it follows from (4.1) that there exists a vertex q ∈ N G (p) with f 0 (q) = f (q) = 2. Thus we may assume that p ∈ {u v : v ∈ L(G)}.For every v ∈ N G (p) ∩ L(G), since f 0 (p) = 0, we have a∈{v,uv}∪V (Qv) f (a) = 3, and hence f (p) = 0, f (v) = 2 and f (z v ) = 2 by Claim 4.1.By the fact that f is a Roman dominating function of H and (4.1), there exists a vertex q ∈ N H (p) − {v,

Theorem 4 . 2
Let l 0 be an integer, and let G be a connected graph of order n ( 3) having exactly l leaves.Then either l = 0 and G

3 .
which contradicts the maximality of |L(H)|.Thus d T (y) = 2 and ∆(T ) Let z be the vertex of T with d T (z) = ∆(T ), and let w ∈ L(T ) be the vertex such that the path P z of T joining w and z contains y.Since d T (z) 3, T 1 = T − (V (P z ) − {z}) is a spider with L(T 1 ) = L(T ) − {w}.

2 . ( 4 . 6 ) 11 ,Remark 3
1.1 leads to γ R (G) 2n − 2l.(4.5)We show that γ R (G) n − l 2 .If l ∈ {0, 1}, then we can easily check that γ R (G) n − 1.Thus we may assume that l 2. Then by Lemma 4.3, G has a subgraph G ′ such that each component of G ′ is a spider and L(G ′ ) = L(G).For each component T of G ′ , let x T be a vertex of T with d T (x T ) = ∆(T ), and letX = {x T : T is a component of G ′ }.Note that |X| l 2 and | x∈X (N G (x) − X)| x∈X |N G ′ (x)| = l.Hence the function f : V (G) → {0, 1, 2} with f ∈ X) 0 (a ∈ x∈X (N G (x) − X)) 1 (Otherwise) is a Roman dominating function of G with w(f ) = n + |X| − | x∈X (N G (x) − X)| n − l 2 .Consequently we have γ R (G) n − l By Fact 1.1, G has a spanning l-leaf minimally connected subgraph H. Since deleting an edge cannot decrease the Roman domination number, we have γ R (G) γ R (H).This together with Theorem 4.1 implies that either l = 0 andG ∈ {B 1 R , B 2 R , B 3 R } or γ R (G) 8n+2l 11 .Since B R is the set of graphs B containing one of B 1 R , B 2 R and B 3R as a spanning subgraph and satisfying γ(B) > 8|V (B)| this together with (4.5) and (4.6) implies that either l = 0 and H ∈ B R or γ(G) min{ 8n+2l 11 , n − l 2 , 2n − 2l}.Consequently Theorem 4.2 holds.In the proof of Theorem 4.2, we further assume that γ R (G) 4n 5 .Then it follows from Theorem 4.1 that l = 2n 5 and H ∈ F 2n 5

l 2 )i l 2 } 5 l n 2 2 n 1 i l 2 } 1 2
, we define the attachment vertex of L i as one of the central vertices of L i .Then every minimal-connected graph G with respect to {L i : 1 has n vertices and l leaves, and satisfies γ R (G) = n − l 2 .Thus Theorem 4.2 for the case where 2n+1 is best possible.Again assume that l is even, and let n be an integer with 3l 2l − 1.For each i (1 i n − 3l2 ), let L ′ i be a path of order 4, and for each i(n − 3l 2 + 1 i l 2 ), let L ′i be a path of order 3.For each i (1 i l 2 ), we define the attachment vertex of L ′ i as one of the central vertices of L ′ i .Then every minimal-connected graph G with respect to {L ′ i : has n vertices and l leaves, and satisfies γ R (G) = n − l 2 .Thus Theorem 4.2 for the case where n+1 2 l 2n 3 is best possible.Let n be an integer with l + 1 n 3l−.Let L ′′ 1 be a star having exactly 3l − 2n + 2 leaves, and for each 2 i n − l, let L ′′ i be a star of order 3.For each i (1 i n − l), we define the attachment vertex of L ′′ i as one of the central vertices of L ′′ i .Then every minimal-connected graph G with respect to {L ′′ i : 1 i n − l} has n vertices and l leaves, and satisfies γ R (G) = 2n − 2l.Thus Theorem 4.2 for the case where 2n+1 3 l n − 1 is best possible.
R t .Furthermore, by Lemma 1.2 and the construction of H, we see that H is 0-leaf minimally connected.This together with Theorem I leads to γ t (H) leads to the desired conclusion.By the construction of H, there exist l disjoint cycles of H having 6 vertices such that G is obtained from H by deleting 4 consecutive vertices of each of those cycles.It follows that G ∈ F l t .This completes the proof of Theorem 3.1.As a corollary of Theorem 3.1, we get the following theorem.
t } ∪ R t , it follows from Theorem I that H ∈ F 0 t .
In the proof of Theorem 3.2, we further assume that γ t (G)