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.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
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[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 \(H\)\()\).
References
Arumugam, S., Jose, B.K., Bujtás, C., Tuza, Z.: Equality of domination and transversal numbers in hypergraphs. Discrete Appl. Math. 161, 1859–1867 (2013)
Chartrand, G., Lesniak, L., Zhang, P.: Graphs and Digraphs. Chapman and Hall/CRC, Boca Raton (2015)
Hartnell, B., Rall, D.F.: A characterization of graphs in which some minimum dominating set covers all the edges. Czechoslov. Math. J. 45, 221–230 (1995)
Lingas, A., Miotk, M., Topp, J., Żyliński, P.: Graphs with equal domination and covering numbers (2018). arXiv:1802.09051v1 [math.CO] (manuscript; 25 Feb 2018)
Randerath, B., Volkmann, L.: Characterization of graphs with equal domination and covering number. Discrete Math. 191, 159–169 (1998)
Wu, Y., Yu, Q.: A characterization of graphs with equal domination number and vertex cover number. Bull. Malays. Math. Sci. Soc. 35, 803–806 (2012)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Miotk, M., Topp, J. & Żyliński, P. Bipartization of Graphs. Graphs and Combinatorics 35, 1169–1177 (2019). https://doi.org/10.1007/s00373-019-02068-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00373-019-02068-5