Bipartization of Graphs

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.

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).

Fig. 1

Graphs H, $B_f(H)$, and $B_g(H)$

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.

Fig. 2

Graph G is the bipartization of the two non-isomorphic graphs H and F

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[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

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$$)$.