Abstract
A dominating set of a graph G is a set \(D\subseteq V_G\) such that every vertex in \(V_GD\) is adjacent to at least one vertex in D, and the domination number \(\gamma (G)\) of G is the minimum cardinality of a dominating set of G. In this paper we provide a new characterization of bipartite graphs whose domination number is equal to the cardinality of its smaller partite set. Our characterization is based upon a new graph operation.
Keywords
Bipartite graph Bipartization Domination numberMathematics Subject Classification
05C69 05C76 05C051 Introduction and Notation
For notation and graph theory terminology we in general follow [2]. Specifically, let \(G=(V_G,E_G)\) be a graph with vertex set \(V_G\) and edge set \(E_G\). For a subset \(X \subseteq ~V_G\), the subgraph induced by X is denoted by G[X]. For simplicity of notation, if \(X=\{x_1,\ldots , x_k\}\), we shall write \(G[x_1,\ldots , x_k]\) instead of \(G[\{x_1,\ldots , x_k\}]\). For a vertex v of G, its neighborhood, denoted by \(N_{G}(v)\), is the set of all vertices adjacent to v, and the cardinality of \(N_G(v)\), denoted by \(\deg _G(v)\), is called the degree of v. The closed neighborhood of v, denoted by \(N_{G}[v]\), is the set \(N_{G}(v)\cup \{v\}\). In general, the neighborhood of \(X \subseteq V_G\), denoted by \(N_{G}(X)\), is defined to be \(\bigcup _{v\in X}N_{G}(v)\), and the closed neighborhood of X, denoted by \(N_{G}[X]\), is the set \(N_{G}(X)\cup X\). A vertex of degree one is called a leaf, and the only neighbor of a leaf is called its support vertex (or simply, its support). A weak support is a vertex adjacent to exactly one leaf. Finally, the set of leaves and the set of supports of G we denoted by \(L_G\) and \(S_G\), respectively.
A subset D of \(V_G\) is said to be a dominating set of a graph G if each vertex belonging to the set \(V_G D\) has a neighbor in D. The cardinality of a minimum dominating set of G is called the domination number of G and is denoted by \(\gamma (G)\). A subset \(C \subseteq V_G\) is a covering set of G if each edge of G has an endvertex in C. The cardinality of a minimum covering set of G is called the covering number of G and denoted by \(\beta (G)\).
It is obvious that if \(G=((A,B),E_G)\) is a connected bipartite graph, then \(\gamma (G)\le \min \{A,B\}\). In this paper the set of all connected bipartite graphs \(G=((A,B),E_G)\) in which \(\gamma (G) = \min \{A,B\}\) is denoted by \(\mathcal{B}\). Some properties of the graphs belonging to the set \(\mathcal{B}\) were observed in the papers [1, 3, 4, 5, 6], where all graphs with the domination number equal to the covering number were characterized. In this paper, inspired by results and constructions of Hartnell and Rall [3], we introduce a new graph operation, called the bipartization of a graph with respect to a function, study basic properties of this operation, and provide a new characterization of the graphs belonging to the set \(\mathcal{B}\) in terms of this new operation.
2 Bipartization of a Graph
Let \(\mathcal{K}_H\) denote the set of all complete subgraphs of a graph H. If \(v\in V_H\), then the set \(\{K\in \mathcal{K}_H :v \in V_K\}\) is denoted by \(\mathcal{K}_H(v)\). If \(X\subseteq V_H\), then the set \(\bigcup _{v\in X}\mathcal{K}_H(v)\) is denoted by \(\mathcal{K}_H(X)\), and it is obvious that \(\mathcal{K}_H(X) = \{K\in \mathcal{K}_H :V_K\cap X\)\(\not =\emptyset \}\). Let \(f:\mathcal{K}_H\rightarrow \mathbb {N}\) be a function. If \(K\in \mathcal{K}_H\), then by \(\mathcal{F}_K\) we denote the set \(\{(K,1),\ldots , (K,f(K))\}\) if \(f(K)\ge 1\), and we let \(\mathcal{F}_K=\emptyset \) if \(f(K)=0\). By \(\mathcal{K}_H^f\) we denote the set of all positively fvalued complete subgraphs of H, that is, \(\mathcal{K}_H^f\)\(=\{K\in \mathcal{{K}}_H:f(K)\ge 1\}\).
Definition 1
Let H be a graph and let \(f:\mathcal{K}_H\rightarrow \mathbb {N}\) be a function. The bipartization of H with respect to f is the bipartite graph \(B_f(H) =((A,B),E_{B_f(H)})\) in which \(A=V_H\), \(B=\bigcup _{K\in \mathcal{K}_H}\mathcal{F}_K\), and where a vertex \(x\in A\) is adjacent to a vertex \((K,i)\in B\) if and only if x is a vertex of the complete graph K\((\)\(i=1,\ldots , f(K)\)\()\).
Example 1
Figure 1 presents a graph H (for which \(\mathcal{K}_H=\{H[a], H[b], H[c], H[d],\)\(H[a, b],H[a,c], H[b,c], H[c,d], H[a,b,c]\}\)) and its two bipartizations \(B_f(H)\) and \(B_g(H)\) with respect to functions \(f,\,g:\mathcal{K}_H\rightarrow \mathbb {N}\), respectively, where \(f(H[a])=1\), \(f(H[b])=1\), \(f(H[c])=2\), \(f(H[d])=0\), \(f(H[a,b])=3\), \(f(H[a,c])=0\), \(f(H[b,c])=2\), \(f(H[c,d])=3\), \(f(H[a, b, c])=1\), while \(g(H[v])=0\) for every vertex \(v \in V_H\), \(g(H[u,v])=1\) for every edge \(uv \in E_H\), and \(g(H[a,b,c]) = 0\). Observe that \(B_g(H)\) is the subdivision graph S(H) of H (i.e., the graph obtained from H by inserting a new vertex into each edge of H).
3 Properties of Bipartizations of Graphs
It is clear from the above definition of the bipartization of a graph with respect to a function that we have the following proposition.
Proposition 1
 (1)
 If \(B_f(H)=((A,B),E_{B_f(H)})\) is the bipartization of a graph H with respect to a function \(f:\mathcal{K}_H\rightarrow \mathbb {N}\), then:
 (a)

\(N_{B_f(H)}(v)= \bigcup _{K\in \mathcal{K}_H(v)}\mathcal{F}_K\) if \(v\in A\).
 (b)

\(N_{B_f(H)}(X)= \bigcup _{K\in \mathcal{K}_H(X)}\mathcal{F}_K\) if \(X\subseteq A\).
 (c)

\(N_{B_f(H)}((K,i))= V_K\) if \((K,i)\in B\)\((\)\(i=1,\ldots , f(K)\)\()\).
 (d)

\(V_{B_f(H)} = V_H + \sum _{K \in \mathcal{K}_H} f(K)\) and \(E_{B_f(H)} = \sum _{K \in \mathcal{K}_H} f(K) \, V_K\).
 (2)

If H is a connected graph and \(f :\mathcal{{K}}_H \rightarrow \mathbb {N}\) is a function such that every edge of H belongs to a positively fvalued complete subgraph of H, then the bipartization \(B_f(H)\) is a connected graph.
 (3)

If H is a graph and \(f,\, g :\mathcal{K}_H\rightarrow \mathbb {N}\) are functions such that \(f(K) \ge g(K)\) for every \(K\in \mathcal{K}_H\), then the graph \(B_g(H)\) is an induced subgraph of \(B_f(H)\).
Our study of properties of bipartizations we begin by showing that every bipartite graph is the bipartization of some graph with respect to some function.
Theorem 1
For every bipartite graph \(G=((A,B),E_G)\) there exist a graph H and a function \(f:\mathcal{K}_H\rightarrow \mathbb {N}\) such that \(G=B_f(H)\).
Proof
We say that vertices x and y of G are similar if \(N_G(x)=N_G(y)\). It is obvious that this similarity is an equivalence relation on B (as well as on A and \(A \cup B\)). Let \(B_1,\ldots , B_l\) be the equivalence classes of this relation on B, say \(B_i = \{b_1^i,b_2^i,\ldots ,b_{k_i}^i\}\) for \(i=1,\ldots , l\). It follows from properties of the equivalence classes that \(B_1+\cdots +B_l= B\), \(N_G(b_1^i)=N_G(x)\) for every \(x\in B_i\), and \(N_G(b_1^i)\not =N_G(b_1^j)\) if \(i, j\in \{ 1,\ldots ,l\}\) and \(i\not =j\).
Now, let \(H=(V_{H},E_{H})\) be a graph in which \(V_H=A\) and two vertices x and y are adjacent in H if and only if they are at distance two apart from each other in G. Let \(\mathcal{K}_H\) be the set of all complete subgraphs of H, and let \(f:\mathcal{K}_H\rightarrow \mathbb {N}\) be a function such that \(f(K)= \{b\in B:N_G(b)=V_K\}\) for \(K\in \mathcal{K}_H\). Next, let \(K_i\) be the induced subgraph \(H[N_G(b_1^i)]\) of H. It follows from the definition of H that \(K_i\) is a complete subgraph of H. In addition, from the definition of f and from properties of the classes \(B_1,\ldots , B_l\), it follows that \(f(K_i)=B_i>0\) (\(i=1,\ldots ,l\)), and \(f(K)=0\) if \(K\in \mathcal{K}_H\{K_1,\ldots , K_l\}\). Consequently, \(\mathcal{K}_H^f=\{ K_1,\ldots ,K_l\}\).
Finally, consider the bipartite graph \(B_f(H) =((X,Y),E_{B_f(H)})\) in which \(X=V_H=A\), \(Y= \bigcup _{K\in \mathcal{K}_H}\mathcal{F}_K = \bigcup _{K\in \mathcal{K}_H^f}\mathcal{F}_K = \bigcup _{i=1}^{l} \{(K_i,1),\ldots ,(K_i,k_i)\}\), and where \(N_{B_f(H)}((K_i,j))\)\(= V_{K_i}=N_G(b_1^i)\) for every \((K_i,j)\in Y\). Now, one can observe that the function \(\varphi :A\cup B\rightarrow X\cup Y\), where \(\varphi (x)=x\) if \(x\in A\), and \(\varphi (b_j^i)= (K_i,j)\) if \(b_j^i\in B\), is an isomorphism between graphs G and \(B_f(H)\). \(\square \)
We have proved that a bipartite graph \(G=((A,B),E_G)\) is the bipartization \(B_f(H)\) of a graph \(H =(V_H,E_H)\) (in which \(V_H=A\) and \(E_H=\{xy:\,x,y\)\(\in A\,\text{ and }\,d_G(x,y)=2\}\)) with respect to a function \(f:\mathcal{K}_H\rightarrow \mathbb {N}\), where \(f(K)= \{b\)\(\in B:N_G(b)=V_K\}\) for \(K\in \mathcal{K}_H\). The same graph G is also the bipartization \(B_g(F)\) of a graph \(F=(V_F,E_F)\) (in which \(V_F=B\) and \(E_F=\{xy:\,x,y \in B\, \text{ and }\,\, d_G(x,y)=2 \}\)) with respect to a function \(g:\mathcal{K}_F\rightarrow \mathbb {N}\), where \(g(K)= \{a\in A:N_G(a)=V_K\}\) for \(K\in \mathcal{K}_F\). Consequently, every bipartite graph may be the bipartization of two nonisomorphic graphs.
Example 2
Figure 2 depicts the bipartite graph G which is the bipartization of the nonisomorphic graphs H and F with respect to functions \(\overline{f}:\mathcal{K}_H \rightarrow \mathbb {N}\) and \(\overline{g}:\mathcal{K}_F\rightarrow \mathbb {N}\), respectively, which nonzero values are displayed in the figure.
It is obvious from Theorem 1 that every tree is a bipartization. We are now interested in providing a simple characterization of graphs H and functions \(f:\mathcal{K}_H \rightarrow \mathbb {N}\) for which the bipartization \(B_f(H)\) is a tree. We begin with the following notation: An alternating sequence of vertices and complete graphs \((v_0,F_1,v_1, \ldots , v_{k1},F_k,v_k)\) is said to be a positively fvalued complete \(v_0v_k\) path if \(v_{i1}v_i\) is an edge in the complete graph \(F_i\) for \(i=1,\ldots , k\). We now have the following two useful lemmas.
Lemma 1
Let H be a connected graph, and let \(f :\mathcal{{K}}_H \rightarrow \mathbb {N}\) be a function. If there are two vertices u and v and two distinct internally vertexdisjoint positively fvalued complete \(uv\) paths in H, then the bipartization \(B_f(H)\) contains a cycle.
Proof
If \((v_0=u, F_1,v_1,\ldots , v_{m1},F_m,v_m=v)\) and \((v_0'=u, F_1', v_1',\ldots , v_{n1}',\)\(F_n',v_n'=v)\) are distinct internally vertexdisjoint positively fvalued complete \(uv\) paths in H, then \((v_0, (F_1,1),v_1,\ldots , v_{m1},(F_m,1), v_m)\) and \((v_0', (F_1',1),\)\(v_1',\ldots , v_{n1}',(F_n',1),v_n')\) are distinct \(uv\) paths in \(B_f(H)\), and so they generate at least one cycle in \(B_f(H)\). \(\square \)
Let us recall first that a maximal connected subgraph without a cutvertex is called a block. A graph H is said to be a block graph if each block of H is a complete graph. The next lemma is probably known, therefore we omit its easy inductive proof.
Lemma 2
If \(\mathcal{S}\) is the set of all blocks of a graph H, then \(\sum \nolimits _{B\in \mathcal{S}}\left( V_B1\right) = V_H1\).
Now we are ready for a characterization of graphs which bipartizations (with respect to some functions) are trees.
Theorem 2
 (1)

\(f(K) \le 1\) for every nontrivial complete subgraph K of H.
 (2)

H is a block graph.
 (3)

For a nontrivial complete subgraph K of H is \(f(K)=1\) if and only if K is a block of H.
Proof
Assume that \(B_f(H)\) is a tree. The statement (1) is obvious, for if there were a nontrivial complete subgraph K of H for which \(f(K)\ge 2\), then for any two vertices u and v belonging to K, the sequence (u, (K, 1), v, (K, 2), u) would be a cycle in \(B_f(H)\).
Suppose now that H is not a block graph. Then there exists a block in H, say B, which is not a complete graph. Thus in B there exists a cycle such that not all its chords belong to B. Let \(C=(v_0,v_1,\ldots , v_l,v_0)\) be a shortest such cycle in B. Then \(l\ge 3\) and we distinguish two cases. If C is chordless, then, by Lemma 1, \(B_f(H)\) contains a cycle. Thus assume that C has a chord. We may assume that \(v_0\) is an endvertex of a chord of C, and then let k be the smallest integer such that \(v_0v_k\) is a chord of C. Now the choice of C implies that the vertices \(v_0, v_1, \ldots , v_k\) are mutually adjacent, and therefore, \(k=2\). Similarly, \(v_0, v_k, \ldots , v_l\) are mutually adjacent, and so we must have \(l=3\). Consequently, \(C=(v_0,v_1,v_2,v_3,v_0)\) and \(v_0v_2\) is the only chord of C. Now it is obvious that there are at least two \(v_0v_2\) positively fvalued complete paths in H. From this and from Lemma 1 it follows that the bipartition \(B_f(H)\) contains a cycle. This contradiction completes the proof of the statement (2).
Let B be a block of H. We have already proved that B is a complete graph. Let \(B'\) be a proper nontrivial complete subgraph of B. To prove (3), it suffices to observe that \(f(B')=0\). On the contrary, suppose that \(f(B')\not =0\). We now choose two distinct vertices v and u belonging to \(B'\), and a vertex w belonging to B but not to \(B'\). This clearly forces that there are at least two \(vu\) positively fvalued complete paths in H. Consequently, by Lemma 1, \(B_f(H)\) contains a cycle, and this contradiction completes the proof of the statement (3).
Assume now that the conditions (1)–(3) are satisfied for H and f. Since endvertices of \(B_f(H)\), corresponding to positively fvalued onevertex complete subgraphs of H, are not important to our study of treelike structure of \(B_f(H)\), we can assume without loss of generality that \(f(H[v])=0\) for every vertex \(v \in V_H\). Consequently, H is a block graph and \(f(K)=1\) for every block K of H, while \(f(K')=0\) for every other complete subgraph \(K'\) of H. It remains to prove that \(B_f(H)\) is a tree. Since \(B_f(H)\) is a connected graph, it suffices to show that \(E_{B_f(H)} = V_{B_f(H)}1\). Let \(\mathcal{S}\) be the set of all blocks of H. Then \(\mathcal{K}_H^f=\mathcal{S}\), \(V_{B_f(H)}= V_H+ \sum _{K\in \mathcal{K}_H^f}f(K)= V_H+\mathcal{S}\), and \(E_{B_f(H)}= \sum _{K\in \mathcal{K}_H^f}f(K)V_K= \sum _{K\in \mathcal{S}}V_K= \sum _{K\in \mathcal{S}}(V_K1)+\mathcal{S}\). Now, since \(\sum _{K\in \mathcal{S}}(V_K1) =V_H1\) (by Lemma 2), we finally have \(E_{B_f(H)}\)\(= (V_{H}1)+\mathcal{S}= (V_{H}+\mathcal{S})1= V_{B_f(H)}1\). \(\square \)
Corollary 1
For every connected graph H, there exists a function \(f :\mathcal{K}_{H} \rightarrow \mathbb {N}\) such that the bipartization \(B_f(H)\) is a tree.
Proof
Let F be a spanning block graph of H and let \(f:\mathcal{K}_F \rightarrow \{0,1\}\) be a function such that \(f(K)=1\) if and only if K is a block of F. Clearly, f satisfies the conditions (1)–(3) of Theorem 2, and so the bipartization \(B_f(H)\) is a tree. \(\square \)
Example 3
Figure 2 shows the tree G which is the bipartization of two block graphs H and F with respect to functions \(\overline{f}\) and \(\overline{g}\), respectively, which nonzero values are listed in the same figure.
4 Graphs Belonging to the Family \(\mathcal{B}\)
In this section, we provide an alternative characterization of all bipartite graphs whose domination number is equal to the cardinality of its smaller partite set, that is, we prove that a connected graph G belongs to the class \(\mathcal{B}\) if and only if G is some bipartization of a graph. For that purpose, we need the following lemma.
Lemma 3
 (1)

\(\gamma (G)=A\).
 (2)

\(\gamma (G)=\beta (G)=A\).
 (3)
 G has the following two properties:
 (a)

Each support vertex of G belonging to B is a weak support and each of its nonleaf neighbors is a support.
 (b)

If x and y are vertices belonging to \(A(L_G\cup S_G)\) and \(d_G(x,y)=2\), then there are at least two vertices \(\overline{x}\) and \(\overline{y}\) in B such that \(N_G(\overline{x})= N_G(\overline{y})= \{x,y\}\).
We are ready to establish our main theorem that provides an alternative characterization of the graphs belonging to \(\mathcal{B}\) in terms of the bipartization of a graph.
Theorem 3
 (1)

If \(uv\in E_H\) and \(f(H[u,v]) =0\), then \(f(H')>0\) for some complete subgraph \(H'\) of H containing the edge uv.
 (2)

If \(uv\in E_H\) and \(f(H[u])=f(H[v])=0\), then \(f(H[u,v])\ge 2\).
Proof
Assume first that \(\gamma (G) = A\). Then G has the properties (3a) and (3b) of Lemma 3. Let \(H=(V_H,E_H)\) be a graph in which \(V_H=A\) and \(E_H=\{xy :x,y\)\(\in A \text { and } d_G(x,y)=2\}\), and let \(f:\mathcal{K}_H \rightarrow \mathbb {N}\) be a function such that \(f(K)= \{x\)\(\in B:N_G(x)=V_K\}\) for each \(K\in \mathcal{K}_H\). Then G is the bipartization \(B_f(H)\) of H with respect to f, as we have shown in the proof of Theorem 1. It is obvious that if \(H=K_1\), then \(\mathcal{K}_H=\{H\}\) and it must be \(f(H)\ge 1\) (as otherwise \(G=B_f(H)\) would be a graph of order one). Thus assume that H is nontrivial. Now it remains to prove that f has the properties (1) and (2).
Let uv be an edge of H such that \(f(H[u,v])=0\). Suppose on the contrary that \(f(H')=0\) for every complete subgraph \(H'\) containing the edge uv. Then the vertices u and v do not share a neighbor in \(B_f(H)=G\), so \(d_G(u,v) > 2\) and uv is not an edge in H, a contradiction. This proves the property (1).
Now let uv be an edge of H such that \(f(H[u])=f(H[v])=0\). From these assumptions it follows that \(d_G(u,v)=2\) and neither u nor v is a support vertex in \(G=B_f(H)\). Now we shall prove that none of the vertices u and v is a leaf in G. First, because \(u, v\in A\) and they have a common neighbor, it follows from the first part of the property (3a) of Lemma 3 that at least one of the vertices u and v is not a leaf in G. Suppose now that exactly one of the vertices u and v is a leaf in G, say u is a leaf. Then it follows from the second part of the property (3a) of Lemma 3 that v is a support vertex in \(G=B_f(H)\) and, therefore, \(f(H[v])>0\), a contradiction. Consequently, both u and v are elements of \(AN_G[L_G]\). Thus, since \(d_G(u,v)=2\), the property (3b) of Lemma 3 implies that there are at least two vertices \({\bar{u}}, \bar{v} \in B\) such that \(N_{G}(\bar{u})=N_{G}(\bar{u})=\{u,v\}\). Therefore \(f(H[u,v])= \{x\in B:N_G(x)=\{u, v\}\}\ge \{\bar{u}, \bar{v}\}=2\) and this proves the property (2).
Assume now that H is a connected graph, and \(f:\mathcal{K}_H \rightarrow \mathbb {N}\) is a nonzero function having the properties (1) and (2). We shall prove that in the bipartization \(B_f(H)=((A,B),E_{B_f(H)})\), where \(A=V_H\) and \(B=\bigcup _{K\in \mathcal{K}_H} \mathcal{F}_K\), is \(A\le B\) and \(\gamma (B_f(H))=A\). This is obvious if H is a graph of order 1. Thus assume that H is a graph of order at least 2. From the property (1) it follows that \(B_f(H)\) is a connected graph. We first prove the inequality \(A\le B\). To prove this, it suffices to show that \(B_f(H)\) has an Asaturating matching. We begin by dividing \(A=V_H\) into two subsets \(V_H^1= \{v\in V_H:f(H[v])\ge 1\}\) and \(V_H^0= \{v\in V_H:f(H[v])=0\}\). It is obvious that the edgeset \(M^1= \{v(H[v],1) :v\in V_H^1\}\) is a \(V_H^1\)saturating matching in \(B_f(H)\). Next, we order the set \(V_H^0\) in an arbitrary way, say \(V_H^0=\{v_1,\ldots ,v_n\}\). Now, depending on this order, we consecutively choose edges \(e_1, \ldots , e_n\) in such a way that \(M^1\cup \{e_1,\ldots , e_i\}\) is a \((V_H^1\cup \{v_1,\ldots ,v_i\})\)saturating matching in \(B_f(H)\).
Assume that we have already chosen a \((V_H^1\cup \{v_1,\ldots ,v_{i1}\})\)saturating matching \(M^1\cup \{e_1,\ldots , e_{i1}\}\) in \(B_f(H)\), and consider the next vertex \(v_i \in V_H^0\). If \(N_{H}(v_i)\cap V_H^0\not =\emptyset \), say \(v_j\in N_{H}(v_i)\cap V_H^0\), then \(f(H[v_j])=0\) and therefore \(f(H[v_i,v_j])\ge 2\) (by the property (2)) and the edge \(e_i= v_i(H[v_i,v_j],1)\) if \(j>i\) (\(e_i= v_i(H[v_i,v_j],2)\) if \(j<i\)) together with \(M^1\cup \{e_1,\ldots , e_{i1}\}\) form a \((V_H^1\cup \{v_1,\ldots ,v_i\})\)saturating matching in \(B_f(H)\). Thus assume that \(N_H(v_i) \subseteq V_H^1\). Let v be a neighbor of \(v_i\) in H. If \(f(H[v_i,v])\ge 1\), then the edge \(e_i= v_i(H[v_i,v],1)\) has the desired property. Finally, if \(f(H[v_i,v])=0\), then \(f(H')>0\) for some complete subgraph \(H'\) of H containing the edge \(v_iv\) (by the property (1)) and in this case the edge \(e_i= v_i(H',1)\) has the desired property (as \(N_H(v_i) \subseteq V_H^1\)). Repeating this procedure as many times as needed, an Asaturating matching in \(B_f(H)\) can be obtained.
To complete the proof, it remains to show that \(\gamma (B_f(H)) = A\). In a standard way, suppose to the contrary that \(\gamma (B_f(H)) < A\). Let D be a minimum dominating set of \(B_f(H)\) with \(D \cap A\) as large as possible. Since \(\gamma (B_f(H))=D\), the inequality \(\gamma (B_f(H))<~A\) implies that \(A  D> D \cap B \ge 1\). In addition, since \(D \cap A\) is as large as possible, the set \(V_H^1\)\((= \{v\in V_H:f(H[v])\ge 1\})\) is a subset of \(D \cap A\), while \(A  D\) is a subset of \(V_H^0\)\((= \{v\in V_H:f(H[v])=0\})\). Now, because \(AD>D\cap B\) and each vertex of \(AD\) has a neighbor in \(D\cap B\), the pigeonhole principle implies that there are two vertices x and y in \(AD\) which are adjacent to the same vertex in \(D\cap B\). Hence, x and y are adjacent in H (by the definition of \(B_f(H)\)). Now, since \(f(H[x])= f(H[y])=0\), the property (2) implies that \(f(H[x,y])\ge 2\). Next, since \(N_{B_f(H)}((H[x,y],1))= N_{B_f(H)}((H[x,y],2))= \{x,y\}\) and \(\{x,y\}\cap D=\emptyset \), the vertices (H[x, y], 1) and (H[x, y], 2) belong to \(D\cap B\). Consequently, it is easy to observe that the set \(D'= (D\{(H[x,y],1), (H[x,y],2)\})\cup \{x,y\}\) is a dominating set of \(B_f(H)\), which is impossible as \(D'=D\) and \(D' \cap A > D \cap A\). This completes the proof. \(\square \)
Example 4
The graph H and the function \(f:\mathcal{K}_H \rightarrow \mathbb {N}\) given in Example 1 have the properties (1) and (2) of Theorem 3 and therefore the bipartization \(B_f(H)\) belongs to the family \(\mathcal{B}\), that is, \(\gamma (B_f(H))=A\), where A is the smaller of two partite sets of \(B_f(H)\) shown in Fig. 1.
The graph F and the function \(\overline{g}\) given in Fig. 2 do not satisfy the condition (2) of Theorem 3. However, the bipartization \(G=B_{\overline{g}}(F)\) is a graph belonging to the family \(\mathcal{B}\) since G is also the bipartization \(B_{\overline{f}}(H)\), with H and \(\overline{f}\) given in Fig. 2 and possessing properties (1) and (2) of Theorem 3.
It is obvious that the complete bipartite graph \(K_{m,n}\) is the bipartization of the complete graph \(K_m\) (resp. \(K_n\)) with respect to the function \(f:\mathcal{K}_{K_m} \rightarrow \{0,n\}\), where \(f(K)=0\) if and only if \(K\in \mathcal{K}_{K_m}\{K_m\}\) (resp. \(g:\mathcal{K}_{K_n} \rightarrow \{0,m\}\), where \(g(K)=0\) if and only if \(K\in \mathcal{K}_{K_n}\{K_n\}\)). It is also evident that if \(\min \{m,n\}\ge 3\), then \(K_{m,n}\) does not belong to the family \(\mathcal{B}\) (as \(\gamma (K_{m,n})=2<\min \{m,n\}\)), and neither \(K_m\) and f nor \(K_n\) and g possess the property (2) of Theorem 3.
Finally, as an immediate consequence of Theorems 2 and 3 we have the following simple characterization of trees in which the domination number is equal to the size of a smaller of its partite sets. All such trees are bipartizations of block graphs.
Corollary 2
 (1)

\(f(K)=1\) if K is a block of H, and \(f(K')=0\) if \(K'\) is a nontrivial complete subgraph of H which is not a block of H.
 (2)

\(\max \{f(H[u]),f(H[v])\}\ge 1\) for every edge uv of H (or, equivalently, the set \(\{v\in V_H:f(H[v])\ge 1\}\) is a covering set of \(H\)\()\).
Notes
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