Graphs and Combinatorics

, Volume 35, Issue 5, pp 1169–1177

# Bipartization of Graphs

• Mateusz Miotk
• Jerzy Topp
• Paweł Żyliński
Open Access
Original Paper

## Abstract

A dominating set of a graph G is a set $$D\subseteq V_G$$ such that every vertex in $$V_G-D$$ 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 number

## Mathematics Subject Classification

05C69 05C76 05C05

## 1 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 end-vertex 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 f-valued 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

The bipartization of a graph with respect to a function has the following properties:
(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 f-valued 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 non-isomorphic graphs.

### Example 2

Figure 2 depicts the bipartite graph G which is the bipartization of the non-isomorphic 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 non-zero 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 $$(v_0,F_1,v_1, \ldots , v_{k-1},F_k,v_k)$$ is said to be a positively f-valued complete $$v_0-v_k$$ path if $$v_{i-1}v_i$$ 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 vertex-disjoint positively f-valued complete $$u-v$$ paths in H, then the bipartization $$B_f(H)$$ contains a cycle.

### Proof

If $$(v_0=u, F_1,v_1,\ldots , v_{m-1},F_m,v_m=v)$$ and $$(v_0'=u, F_1', v_1',\ldots , v_{n-1}',$$$$F_n',v_n'=v)$$ are distinct internally vertex-disjoint positively f-valued complete $$u-v$$ paths in H, then $$(v_0, (F_1,1),v_1,\ldots , v_{m-1},(F_m,1), v_m)$$ and $$(v_0', (F_1',1),$$$$v_1',\ldots , v_{n-1}',(F_n',1),v_n')$$ are distinct $$u-v$$ 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 $$\sum \nolimits _{B\in \mathcal{S}}\left( |V_B|-1\right) = |V_H|-1$$.

Now we are ready for a characterization of graphs which bipartizations (with respect to some functions) are trees.

### Theorem 2

Let H be a connected graph, and let $$f :\mathcal{{K}}_H \rightarrow \mathbb {N}$$ be a function such that every edge of H belongs to some positively f-valued complete subgraph of H. Then the bipartization $$B_f(H)$$ is a tree if and only if the following conditions hold:
(1)

$$f(K) \le 1$$ for every non-trivial complete subgraph K of H.

(2)

H is a block graph.

(3)

For a non-trivial 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 non-trivial 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 end-vertex 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 $$v_0-v_2$$ positively f-valued 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 non-trivial 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 $$v-u$$ positively f-valued 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 end-vertices of $$B_f(H)$$, corresponding to positively f-valued one-vertex complete subgraphs of H, are not important to our study of tree-like 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 $$|E_{B_f(H)}|= \sum _{K\in \mathcal{K}_H^f}f(K)|V_K|= \sum _{K\in \mathcal{S}}|V_K|= \sum _{K\in \mathcal{S}}(|V_K|-1)+|\mathcal{S}|$$. Now, since $$\sum _{K\in \mathcal{S}}(|V_K|-1) =|V_H|-1$$ (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 non-zero 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

[4] Let $$G=((A,B),E_G)$$ be a connected bipartite graph with $$1\le |A| \le |B|$$. Then the following statements are equivalent:
(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 non-leaf 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

Let $$G=((A,B),E_G)$$ be a connected bipartite graph with $$1\le |A| \le |B|$$. Then $$\gamma (G) =|A|$$ if and only if G is the bipartization $$B_f(H)$$ of a connected graph H with respect to a non-zero function $$f:\mathcal{K}_H \rightarrow \mathbb {N}$$ and f has the following two properties:
(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 non-trivial. 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 $$A-N_G[L_G]$$. 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 non-zero 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 A-saturating 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 edge-set $$M^1= \{v(H[v],1) :v\in V_H^1\}$$ 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 $$(V_H^1\cup \{v_1,\ldots ,v_{i-1}\})$$-saturating matching $$M^1\cup \{e_1,\ldots , e_{i-1}\}$$ 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 $$M^1\cup \{e_1,\ldots , e_{i-1}\}$$ 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 A-saturating 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 $$|A-D|>|D\cap B|$$ and each vertex of $$A-D$$ has a neighbor in $$D\cap B$$, the pigeonhole principle implies that there are two vertices x and y in $$A-D$$ 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[xy], 1) and (H[xy], 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

Let $$T=((A,B),E_T)$$ be a tree in which $$1\le |A|\le |B|$$. Then $$\gamma (T) =|A|$$ if and only if T is the bipartization $$B_f(H)$$ of a block graph H with respect to a non-zero function $$f:\mathcal{K}_H \rightarrow \mathbb {N}$$ and f has the following two properties:
(1)

$$f(K)=1$$ if K is a block of H, and $$f(K')=0$$ if $$K'$$ is a non-trivial 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 $$H$$$$)$$.

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