Structural Properties of Connected Domination Critical Graphs

A graph $G$ is said to be $k$-$\gamma_{c}$-critical if the connected domination number $\gamma_{c}(G)$ is equal to $k$ and $\gamma_{c}(G + uv)<k$ for any pair of non-adjacent vertices $u$ and $v$ of $G$. Let $\zeta$ be the number of cut vertices of $G$ and let $\zeta_{0}$ be the maximum number of cut vertices that can be contained in one block. For an integer $\ell \geq 0$, a graph $G$ is $\ell$-factor critical if $G - S$ has a perfect matching for any subset $S$ of vertices of size $\ell$. It was proved that, for $k \geq 3$, every $k$-$\gamma_{c}$-critical graph has at most $k - 2$ cut vertices and the graphs with maximum number of cut vertices were characterized. It was proved further that, for $k \geq 4$, every $k$-$\gamma_{c}$-critical graphs satisfies the inequality $\zeta_{0}(G) \le \min \left\{ \left\lfloor \frac{k + 2}{3} \right\rfloor, \zeta \right\}$. In this paper, we characterize all $k$-$\gamma_{c}$-critical graphs having $k - 3$ cut vertices. Further, we establish realizability that, for given $k \geq 4$, $2 \leq \zeta \leq k - 2$ and $2 \leq \zeta_{0} \le \min \left\{ \left\lfloor \frac{k + 2}{3} \right\rfloor, \zeta \right\}$, there exists a $k$-$\gamma_{c}$-critical graph with $\zeta$ cut vertices having a block which contains $\zeta_{0}$ cut vertices. Finally, we proved that every $k$-$\gamma_{c}$-critical graph of odd order with minimum degree two is $1$-factor critical if and only if $1 \leq k \leq 2$. Further, we proved that every $k$-$\gamma_{c}$-critical $K_{1, 3}$-free graph of even order with minimum degree three is $2$-factor critical if and only if $1 \leq k \leq 2$.


Introduction
For a natural number n, we let [n] = {1, 2, ..., n}. All graphs in this paper are finite, undirected and simple (no loops or multiple edges). For a graph G, let V (G) denote the set of vertices of G and let The degree deg(x) of a vertex x in G is |N G (x)|. The minimum degree of G is denoted by δ(G). When no ambiguity occur, we write N (x), N (X) and δ(G) instead of N G (x), N G (X) and δ, respectively. An end vertex is a vertex of degree one and a support vertex is the vertex which is adjacent to an end vertex. A tree is a connected graph with no cycle. A star K 1,n is a tree containing one support vertex and n end vertices. The support vertex of a star is called the center. For a connected graph G, a vertex subset S of G is called a cut set if G − S is not connected. We let ω o (G − S) be the number of components of G − S containing odd number of vertices. In particular, if S = {v}, then v is called a cut vertex of G. That is, G − v is not connected. A block B of a graph G is a maximal connected subgraph such that B has no cut vertex. An end block of G is a block containing exactly one cut vertex of G. For graphs H and G, a graph G is said to be H-free if G does not contain H as an induced subgraph. For a connected graph G, a bridge xy of G is an edge such that G − xy is not connected.
For a finite sequence of graphs G 1 , ..., G l for l ≥ 2, the joins G 1 ∨ · · · ∨ G l is the graph consisting of the disjoint union of G 1 , ..., G l and joining edges from every vertex of G i to every vertex of G i+1 for 1 ≤ i ≤ l − 1. In particular, for a subgraph H of G 2 , the join G 1 ∨ H G 2 is the graph consisting of the disjoint union of G 1 and G 2 and joining edges from every vertex of G 1 to every vertex of H. As the join operation is run over vertices, for a vertex x and a set X of vertices, the join x ∨ X is the graph consisting of the disjoint union of {x} and X and joining edges from x to every vertex in X. The distance d(u, v) between vertices u and v of G is the length of a shortest (u, v)-path in G. The diameter of G diam(G) is the maximum distance of any two vertices of G. For a non-negative integer k, a graph G is k-diameter critical if diam(G) = k and diam(G − uv) > k for any edge uv ∈ E(G). A matching of a graph G is a set of edges which are not incident to a common vertex. A matching M of a graph G is perfect if V (M ) = V (G). For a non-negative integer ℓ, a graph G is ℓ-factor critical if G − S has a perfect matching where S is any set of ℓ vertices, in particular, a graph G is factor critical if ℓ = 1 and is bi-critical if ℓ = 2. For subsets D and X of V (G), D dominates X if every vertex in X is either in D or adjacent to a vertex in D. If D dominates X, then we write D ≻ X and we also write a ≻ X when D = {a}. Moreover, if X = V (G), then D is a dominating set of G and we write D ≻ G instead of D ≻ V (G). A connected dominating set of a graph G is a dominating set D of G such that G[D] is connected. If D is a connected dominating set of G, we then write D ≻ c G. A smallest connected dominating set is called a γ c -set. The cardinality of a γ c -set is called the connected domination number of G and is denoted by γ c (G). A graph G is said to be k-γ c -critical if γ c (G) = k and γ c (G + uv) < k for any pair of non-adjacent vertices u and v of G.
In the structural characterizations of k-γ c -critical graphs, Chen et al. [8] showed that every 1-γ ccritical graph is a complete graph while every 2-γ c -critical graph is the complement of the disjoint union of at least two stars. However, for k = 3, it turns out that the k-γ c -critical graphs have no complete characterization in the sense of free graphs (see [16], Chapter 5). Interestingly, it was proved by Hanson and Wang [10] that, for a connected graph G, the graph G is 3-γ c -critical if and only if the complement of G is 2-diameter critical. For a study on k-diameter critical graphs see Almalki [1]. In [2], Ananchuen proved that every 3-γ c -critical graph contains at most one cut vertex and also established characterizations of 3-γ c -critical graphs having a cut vertex. For more studies related with 3-γ c -critical graphs see [5,20,21]. For k = 4, Kaemawichanurat and Ananchuen [13] proved that every 4-γ c -critical graph contains at most two cut vertices and the characterization of the such graphs having two cut vertices was given. Further, Kaemawichanurat and Ananchuen [14] established that every k-γ c -critical graphs contains at most k − 2 cut vertices when k ≥ 5. They also characterized that there is exactly one class of k-γ c -critical graphs having k − 2 cut vertices. In the same paper, the authors established the maximum number of cut vertices that every block of the graph can have. That is: 14]) Let G be a k-γ c -critical graph containing ζ cut vertices and let ζ 0 (G) be the maximum number of cut vertices of G that can be in a block of G. Then Very recently, Henning and Kaemawichanurat [12] characterized all the eleven classes of k-γ c -critical graphs satisfying the upper bound of Theorem 1.
In this paper, for k ≥ 5, we characterize all k-γ c -critical graphs having k − 3 cut vertices. Further, we establish realizability that, for given k ≥ 4, 2 ≤ ζ ≤ k − 2 and 2 ≤ ζ 0 ≤ min k+2 3 , ζ , there exists a k-γ c -critical graph with ζ cut vertices having a block which contains ζ 0 cut vertices. We also proved that every k-γ c -critical graph of odd order with δ ≥ 2 is factor critical if and only if 1 ≤ k ≤ 2. We prove that every k-γ c -critical K 1,3 -free graph of even order with δ ≥ 2 is factor critical if and only if 1 ≤ k ≤ 2. All the main results are shown in Section 2 while their proofs are given in Section 4. We present some useful results that are used in the proofs in Section 3.
For a k − 3 tuples i = (0, 0, ..., i l , ..., 0) where i l = 1 and i l ′ = 0 for 1 ≤ l = l ′ ≤ k − 2, a graph G in the class G 1 i can be constructed from paths c 0 , c 1 , ..., c l−1 and c l , c l+1 , ..., c k−4 , a copy of a complete graph K n l and a block B ∈ B 2,2 (the construction of the class B 2,2 will be given in Section 3 as it was established earlier in [14]) by adding edges according the join operations : where we call c the head of B. Examples of graphs in this case are illustrated by Figures 1 and 2. Further, for a k − 3 tuples i = (0, 0, ..., 1) where i k−3 = 1 and i l ′ = 0 for 1 ≤ l ′ ≤ k − 2, a graph G in the class G 1 i can be constructed from paths c 0 , c 1 , ..., c k−4 , a copy of a complete graph K n k−3 and a block B ∈ B 2,2 by adding edges according the join operation : where we call c the head of B. Example of a graph in this case is illustrated by Figure 3.  Next, we will construct another class of k-γ c -critical graphs with k − 3 cut vertices. Before giving the construction, we introduce the following class of end blocks.

The class B 3
An end block B ∈ B 3 has b as the head.
Moreover, B has the following properties (2) For every non-adjacent vertices x and y ofB, there exists a γ c -set D B xy of B + xy such that It is worth noting that D v in the property (1) satisfies D v ∩ A = ∅ in order to dominate b. We are ready to give the construction.
The class G 2 (k) For k ≥ 5, a graph G in this class can be constructed from a path c 0 , c 1 , ..., c k−4 and an end block B ∈ B 3 with the head b by adding the edge c k−4 b. For the sake of convenience, we may relabel b as c k−3 .
Then, we let Z(k, ζ) : the class of k-γ c -critical graphs containing ζ cut vertices.
Our first main result is: For our next main results, we let B be a block of G. We further define the following notation. We let C(G) be the set of all cut vertices of G. That is We prove that : , ζ , there exists a k-γ c -critical graph with ζ cut vertices having a block that contains ζ 0 cut vertices.
Finally, we establish a constructive proofs to show that: Theorem 4 Every k-γ c -critical graph of odd order with δ ≥ 2 is factor critical if and only if k ∈ [2].
Theorem 5 Every k-γ c -critical K 1,3 -free graph of even order with δ ≥ 3 is bi-critical if and only if k ∈ [2].

Preliminaries
In this section, we state a number of results that are used in establishing our theorems. We begin with a result of Favaron [9] which gives matching properties of graphs according to the toughness.
Theorem 6 [9] For an integer ℓ ≥ 0, let G be a graph with minimum degree δ ≥ ℓ + 1. Then G is ℓ-factor critical if and only if ω o (G − S) ≤ |S| − ℓ for any cut set S of G such that |S| ≥ ℓ.
In the context of k-γ c -critical graphs, Ananchuen et al. [4] established some matching properties of the such graphs when k = 3.
However, for factor criticality, they found a 3-γ c -critical graph of odd order with δ ≥ 2 which is not factor critical. The graph is constructed from a complete graph K n , a star K 1,n by joining every end vertex of the star to every vertex of K n and then remove all edges of a perfect matching between these two graphs. The resulting graph is detailed in Figure 4.  Figure 4: A 3-γ c -critical graph which is non-factor critical Chen et al. [8] characterized all k-γ c -critical graphs when 1 ≤ k ≤ 2.
We then obtain the following observations as a consequence of theorem 8.
Observation 1 For k ∈ [2] every k-γ c -critical graph of odd order with minimum degree δ ≥ 2 is factor critical.
Observation 2 For k ∈ [2] every k-γ c -critical graph of even order with minimum degree δ ≥ 3 is bi-critical.
By Theorem 8, we observe further that a k-γ c -critical graph does not contain a cut vertex when 1 ≤ k ≤ 2.
Observation 3 Let G be a k-γ c -critical graph with 1 ≤ k ≤ 2. Then G has no cut vertex.
Further, Chen et al. [8] established fundamental properties of k-γ c -critical graphs for k ≥ 2.
Lemma 1 [8] Let G be a k-γ c -critical graph, x and y a pair of non-adjacent vertices of G and D xy a γ c -set of G + xy. Then Ananchuen [2] established structures of k-γ c -critical with a cut vertex. All the following results of this section were established in [14]. The first result is the construction of a forbidden subgraph of k-γ c -critical graphs. For a connected graph G, let X, Y, X 1 and Y 1 be disjoint vertex subsets of V (G). The induced subgraph An example of a bad subgraph is illustrated in Figure 5.
The authors showed, in [14], that : Lemma 3 [14] For k ≥ 3, let G be a k-γ c -critical graph. Then G does not contain a bad subgraph.
They also provided characterizations of some blocks of k-γ c -critical graphs. Recall that C(G) is the set of all cut vertices of G and, for a block B of G, Figure 5: When no ambiguity occur, we write C rather than C(G). In the same paper, the authors showed further that for a connected graph G and a pair of non-adjacent vertices x and y of G, C(G) = C(G + xy) if x and y are in the same block of G.

Lemma 4 [14]
For a connected graph G, let B be a block of G and x, y ∈ V (B) such that xy / ∈ E(G). Then C(G) = C(G + xy).
Let D be a γ c -set of G. The followings are the characterization of four classes of end blocks of k-γ c -critical graphs that contains at most 3 vertices from D. For vertices c, z 1 and z 2 , let B 0 = {c ∨ K t1 : for an integer t 1 ≥ 1}, The following is a part of the construction of B 2,2 . For integers l ≥ 2, m i ≥ 1 and r ≥ 0, we let S = ∪ l i=1 K 1,mi and T = S ∪ K r . When r = 0, we let T = S. Then, for That is, T can be obtained by removing the edges in the stars of S from a complete graph on S∪S ′ ∪S ′′ . Then the blocks in B 2,2 are defined as follows.
A graph in this class is illustrated by Figure 6. According to the figure, an oval denotes a complete subgraph, double lines between subgraphs denote all possible edges between the subgraphs and a dash line denotes a removed edge.
The following are the characterizations of an end block B such that |D ∩ V (B)| ≤ 3.
Lemma 6 [14] Let G be a k-γ c -critical graph with a γ c -set D and let B be an end block of G.
Lemma 7 [14] Let G be a k-γ c -critical graph with a γ c -set D and let B be an end block of G.
Finally, we conclude this section by the following two lemmas which are structures of blocks of k-γ ccritical graphs. Recall from Section 2 that Z(k, ζ) : the class of k-γ c -critical graphs containing ζ cut vertices.
Then G has only two end blocks and another blocks contain exactly two cut vertices.
Lemma 10 [14] Let G ∈ Z(k, ζ) with a γ c -set D where ζ ∈ {k − 3, k − 2} and B be a block of G containing two cut vertices c and c ′ .
In [11], the authors established a construction of the class P(k) which, for any graph G ∈ P(k) and integer l ≥ 1, there exists a (k + l)-γ c -critical graph contains G as an induced subgraph. Recall that, for a subgraph H of G, H is maximal complete subgraph of G if for any complete subgraph

The class P(k)
A k-γ c -critical graph G is in this class if there exists a maximal complete subgraph H of order at least two of G satisfies the following properties : We next give a construction of a (k + l)-γ c -critical graph containing G in the class P(k) as an induced subgraph. Let H be a maximal complete subgraph of G having properties (i) and (ii). The graph G(n 1 , n 2 , ..., n l ) can be constructed from a vertex x 0 , l copies of completes graph K n1 , K n2 , ..., K n l−1 which n i ≥ 1 for 1 ≤ i ≤ l and the graph G by adding edges according to the join operations : The graph is illustrated by Figure 7. Thus, they proved that Theorem 9 [11] For an integer k ≥ 3, let G ∈ P(k). Then G(n 1 , n 2 , ..., n l ) is a (k + l)-γ c -critical graph for all l ≥ 1.

Connected Dominating Set of Blocks
Let B(G) be the family of all blocks of G.
When no ambiguity can occur, we use B to denote B(G). For a k-γ c -critical graph G with a cut vertex, let B be an end block of G containing non-adjacent vertices x and y. Clearly, So c ∈ D ∩C and thus, D xy ∩C ⊆ D ∩C, as required.
This completes the proof. ✷ It is worth noting that, in [14], the similar result as Lemma 11 was proved but focused only end blocks. So, our result in Lemma 11 is more general. For non-adjacent vertices x and y of the block B, the following lemma gives the number of vertices of a γ c -set of G + xy in B.
Proof. We first establish the following claim.
contradicting the minimality of D. Thus establishing the claim.
We now prove Lemma 12. Suppose to the contrary that This contradicts Lemma 1(1). Thus |D xy ∩ V (B)| < |D ∩ V (B)| and this completes the proof. ✷ In this subsection, we characterize k-γ c -critical graphs with k − 3 cut vertices. We recall the classes G 1 (i 1 , i 2 , ..., i k−3 ) and G 2 (k) from Section 2. First, we will prove that all graphs in these two classes are k-γ c -critical with k − 3 cut vertices.
Proof. Clearly G has c 1 , c 2 , ..., c k−4 and c as the k − 3 cut vertices. Observe that, for any i = To prove all cases of i , we may relabel the path P to be x 1 , ..., Now, we will establish the criticality. Let u and v be a pair of non-adjacent vertices of G and let S 1 = S ∪ S ′ ∪ S ′′ . We first assume that |{u, v} ∩ S 1 | = 0. Therefore {u, v} ⊆ {x 1 , x 2 , ..., x k−2 , x k−1 }.

Lemma 14
Let G be a graph in the class G 2 (k). Then G is a k-γ c -critical graph with k − 3 cut vertices. In the following, we let G ∈ Z(k, k − 3) having a γ c -set D. In view of Lemma 9, G has only two end blocks and another blocks contain two cut vertices. Thus, we let B 1 and B k−2 be the two end blocks and another blocks B 2 , B 3 , ..., B k−3 contain two cut vertices. Without loss of generality let Proof. Suppose to the contrary that |V (C 1 ) ∩ D| ≤ 1 and |V (C k−2 ) ∩ D| ≤ 1. Lemmas 5 and 6 imply that B 1 , B k−2 ∈ B 0 ∪B 1 . This contradicts Lemma 8. Thus either |V (C 1 )∩D| ≥ 2 or |V (C k−2 )∩D| ≥ 2 and this completes the proof. ✷ By Lemma 15, we may suppose without loss of generality that |V (C k−2 ) ∩ D| ≥ |V (C 1 ) ∩ D|.
Proof. As G ∈ H(b 1 , b 2 , ..., b k−3 , 2) and b i = 1 for some 1 ≤ i ≤ k − 3, we must have b j = 0 for all 1 ≤ j = i ≤ k − 3. Because B i contains two cut vertices, i > 1. Therefore, b 1 = 0. Lemma 5 then implies that We first show that B ′ is complete. Let x and y be non-adjacent vertices of B ′ . Consider G + xy. Lemma 1(2) implies that |D xy ∩ {x, y}| ≥ 1. As x, y ∈ V (B ′ ), we must have We will show that c i−1 c i / ∈ E(G). Hence, we may assume to the contrary that c i−1 c i ∈ E(G). We let Since |D ∩ (V (B i ) − {c i−1 , c i })| = 1, it follows that X = ∅. Because B ′ is complete, G[X 1 ∪ X] is complete. In fact, X 1 and X satisfy (i) and (ii) of bad subgraphs. We then let Thus G has X, X, Y and Y 1 as a bad subgraph. This contradicts Lemma 3. Hence, c i−1 c i / ∈ E(G).

We finally show that
. We may assume to the contrary that there exists a vertex u of B ′ which is not adjacent to c i−1 . Consider G + uc i−1 . Corollary 1 gives that is not connected, a contradiction. Hence, N Bi (c i−1 ) = V (B ′ ) and, similarly, N Bi (c i ) = V (B ′ ). Since B i is a block, n i ≥ 2 and this completes the proof. ✷ We will prove the following two theorems, both of which give main contribution to the proof of our first main theorem, Theorem 2.
We will show that B k−2 ∈ B 2,2 . Clearly, b 1 is either 0 or 1. If b 1 = 0, then Lemma 5 implies that where z 2 is given at the definition of B 2,1 . We then let Clearly, G has a bad subgraph, contradicting Lemma 3. Thus B k−2 ∈ B 2,2 .
As b i = 0 for all 2 ≤ i ≤ k − 3, by Lemma 10, B i = c i−1 c i . By Lemma 5 and similar arguments in Theorem 10, we have that B 1 = c 0 c 1 .
To dominate c 1 , we have that c 2 ∈ D c0v . By the connectedness of (G (1). This completes the proof of Case 2.

k-γ c -Critical Graphs with Prescribed Cut Vertices
In this section, we prove Theorem 3. First, we introduce structure of some subgraphs. For any block H of a graph G, H is call a block H ℓ for ℓ ≥ 2 if H consists of a vertex x and U 1 , U 2 , ..., U ℓ as vertex sets of order at least 2 and We say that x is the head of a block H ℓ .

Let
D(k, ζ, ζ 0 ) : the class of all k-γ c -critical graphs with ζ cut vertices containing a block B such that ζ(B) = ζ 0 . We next introduce the following class that we use to establish the existence of graphs in D(k, ζ, ζ 0 ).
The class F (p, q, r): Let H 1 , H 2 , ..., H p be p of H 2 blocks. Further, we let H p+1 be P q (a path of q vertices), and let H p+2 be an H r block. Let c i be the head of H i for 1 ≤ i ≤ p + 2. A graph G ∈ F (p, q, r) is obtained from H 1 , ..., H p+2 by joining edges between vertices in {c i : 1 ≤ i ≤ p + 2} to form a clique. F (p, q, r), then G is (r + q + 3p)-γ c -critical with p + q cut vertices having a block that contains p + 2 cut vertices.

Lemma 18 If a graph G is in the class
We see that U i 1 and U i 2 are, respectively, the same as U 1 and U 2 of the block H 2 . Let a i ∈ U i 1 . By the construction of the block For the block H p+2 , letŨ i be a vertex subset of V (H p+2 ) which is the same as U i of the block H r and u i ∈Ũ i for i ∈ [r].
Let k = r + q + 3p. For a pair of non adjacent vertices u, v ∈ V (G). Consider G + uv. To establish the criticality, it suffices to show that there exists a dominating set D uv of G + uv containing less than k vertices. For i ∈ [p + 2], let D i = D ∩ V (H i ). We distinguish 3 cases.
For j ≥ 2, We now consider the case when v ∈ V (H p+1 ). We let Subcase 1.2 : u ∈ U 1 1 . Without loss of generality let u = a 1 . So ua ′ 1 / ∈ E(G) and ub 1 ∈ E(G). Clearly, v = c 1 . If v = c i for some {2, ..., p}, then we can find D uv by the same arguments as Subcase 1.1. Thus, we may consider when v = c i for i = 1, 2, ..., p.
We now consider the case We letD = D uv = (∪ p i=2 D i ) ∪ {c 1 , u, a 1 } and we can find D uv by same arguments as Subcase 1.2.
. By the same arguments as Finally, we see that c 1 , ..., c p+2 , d 1 , ..., d q−2 are all the cut vertices of G. Thus, G has p + q cut vertices. Further, the block G[{c 1 , ..., c p+2 }] has c 1 , ..., c p+2 as the cut vertices of G. Therefore, there is a block containing p + 2 cut vertices. This completes the proof. ✷ Now, we are ready to prove Theorem 3. For completeness, we restate the theorem.

Factor Criticality of k-γ c -Critical Graphs
In this section, for k ≥ 3, we will use the property of graphs in the class P(k) which is given in Section 3. First, we may prove that the class P(k) is non-empty for k ≥ 3.
Proof. For an integer k ≥ 3, we let C k+2 = c 1 , c 2 , ..., c k+2 , c 1 be a cycle of length k + 2. It is well known that C k+2 is a k-γ c -critical graph. In this proof, all subscripts are taken modulo k + 2. Observe that C k+2 [{c j , c j+1 }] is a maximal complete subgraph for all 1 ≤ j ≤ k + 2. Thus, without loss of generality, it suffices to show that complete subgraph C k+2 [{c 1 , c 2 }] satisfies (i) and (ii) of P(k).
As k + 2 ≥ 5, at least one of C 1 or C 2 must have at least three vertices. Without loss of generality let it be C 1 . We further let . Hence C k+2 ∈ P(k) and this completes the proof. ✷ In the following, we show how to apply the construction of some graphs in the class P(k) to establish the existence of (k + l)-γ c -critical graphs with some property. For an integer k ≥ 1 and ℓ ≥ 1, we let Q(k, ℓ) : the class of k-γ c -critical graphs G with δ ≥ ℓ + 1 such that G is not ℓ-factor critical.
Hence, we may rewrite Observation 1 in term of the class Q(k, 1).
For k = 3, Figure 4 shows that there exists a 3-γ c -critical graph of odd order and δ ≥ 2 which is non-factor critical. Thus Q(3, 1) = ∅. In the following, for k ≥ 4, we show further that there exists a k-γ c -critical graph which is non-factor critical.
The class X (s) For an integer s ≥ 3, let A = {a 1 , a 2 , ..., a s } and B = {b 1 , b 2 , ..., b s } be two disjoint sets of vertices. We further let K s be a copy of a complete graph of order s such that V (K s ) = {y 1 , y 2 , ..., y s }. A graph G in the class X (s) can be constructed from A, B and K s by adding edges according to the join operations that, for 1 ≤ i ≤ s, A graph in this class is illustrated by Figure 8.   The following lemma gives that X (s) ⊆ P(4) ∩ Q(4, 1) for integer s ≥ 3.
Proof. For a given s ≥ 3, let G be in the class X (s). We first show that γ c (G) = 4. Suppose to the contrary that there exists a connected dominating set D of size less than 4. We first consider the case when D ∩ V (K s ) = ∅. To dominate B, |D ∩ A| ≥ 2. Therefore D = {y i , a j , a l }. By the connectedness of G[D], i / ∈ {j, l}. Hence D does not dominate a i , a contradiction. Thus, we consider the case when D ∩ V (K s ) = ∅. So, to dominate K s , |D ∩ A| ≥ 2. Since A is an independent set, by the connectedness of G[D], |D ∩ B| ≥ 1. Therefore D = {a j , a l , b i }. Similarly i / ∈ {j, l} and this implies that D does not dominate a i , a contradiction. Thus γ c (G) ≥ 4.
We observe that K s is a maximal complete subgraph of G. We do not only show that γ c (G) ≤ 4 but we also show that, for a vertex a of G, there exists a connected dominating set D a of G containing a and D ∩ V (K s ) = ∅. That is we show that K s satisfies (i) of graphs in the class P(k). For Hence γ c (G) ≤ 4 and thus γ c (G) = 4. Moreover, K s satisfies (i).
We finally establish the criticality. Further, we show that, for non-adjacent vertices x and y of G, D xy ∩ V (K s ) = ∅. That is we will show that K s satisfies (ii) of graphs in the class P(k). We first consider the case when {x, y} ∩ B = ∅. If {x, y} = {b i , b j }, then D xy = {b i , a j , y l }. If {x, y} = {a i , b i }, then D xy = {a i , y j , y l }. If {x, y} = {b i , y j } (in this case y j could be y i ), then D xy = {y j , y l , a i }. We now consider the case when {x, y} ∩ B = ∅. Thus {x, y} is either {a i , a j } or {a i , y i }. In both cases, D xy = {a i , a j , y i }. Thus G is a 4-γ c -critical graph, in particular, G ∈ P(4).
Finally, let s be odd number and S = A. Thus, ω o (G−S) has K s and b 1 , ..., b s as s+1 odd components.
Thus ω o (G − S) = s + 1 > s − 1 = |S| − 1. By Theorem 6, G is non-factor critical. Thus, G ∈ Q(4, 1) and this completes the proof. ✷ We will use a graph in the class P(4) to show that Q(k, 1) = ∅ for all k ≥ 6. For k = 5, we also provide a graph G 5 (l 1 , l 2 ) in the class Q(5, 1) by the following construction. Let u, x, y, z and w be five different vertices. We also let P 2 = x ′ , y ′ be a path of length one and K l1 , K l2 be two copies of complete graphs of order l 1 ≥ 2 and l 2 ≥ 2 respectively, moreover, l 1 + l 2 is even number. The graph G 5 (l 1 , l 2 ) is constructed by adding edges according to the join operations : • z ∨ {x, y, w}. It is not difficult to show that G 5 (l 1 , l 2 ) is 5-γ c -critical graph. Moreover, G 5 (l 1 , l 2 ) has S = {x ′ , y ′ , z} as a cut set such that ω o (G 5 (l 1 , l 2 ) − S) = 4 = |S| + 1 > |S| − 1. By Theorem 6, G 5 (l 1 , l 2 ) is not factor critical. Now, we are ready to prove Theorem 4. For completeness, we restate the theorem. Proof. Observation 4 implies that if k = 1 or 2, then Q(k, 1) = ∅.
Theorem 6 then gives that G(n 1 , n 2 , ..., n l−1 , 1) is non-factor critical. Therefore Q(k, 1) = ∅ for all k ≥ 6. These imply that if Q(k, 1) = ∅, then k = 1 or 2. This completes the proof. ✷ In view of Theorem 4, it is natural to think of the bi-criticality of k-γ c -critical graphs with δ ≥ 3. Although, we know that if a graph G is not factor critical, then there exists a cut set S such that By Theorem 6, regardless with the parity of the orders of graphs, it is most likely there exist k-γ ccritical graphs that are not bi-critical. However, we notice that the graphs that are obtained from the construction in Figure 9 and Theorem 4 contain a claw, K 1,3 , as an induced subgraphs. Hence, we may ask if every k-γ c -critical K 1,3 -free graph with δ ≥ 3 is bi-critical. By Theorem 8, for 1 ≤ k ≤ 2, every k-γ c -critical graph is K 1,3 -free with δ ≥ 3 of even order is bi-critical. Therefore, we obtain the following corollary by Observation 2 Corollary 2 If k ∈ [2], then every k-γ c -critical K 1,3 -free graph with δ ≥ 3 is bi-critical.
When k ≥ 3, it turns out that there exist k-γ c -critical graph with δ ≥ 3 which is not bi-critical even they do not contain K 1,3 as an induced subgraph. LetQ (k, ℓ) : the class of k-γ c -critical K 1,3 -free graphs G with δ ≥ ℓ + 1 such that G is not ℓ-factor critical.
The class A(t 1 , t 2 ) For an odd number t 1 ≥ 2 and an even number t 2 ≥ 2, the graph G in this class is obtained from vertices x 1 , x 2 , x 3 and copies of complete graphs K t1 , K t2 by adding edges according to the join operations: