Bipartization of graphs

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.


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 ⊆ V G , the subgraph induced by X is denoted by G[X]. For simplicity of notation, if X = {x 1 , . . . , x k }, we shall write G[x 1 , . . . , x k ] instead of G[{x 1 , . . . , 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) ∪ {v}. In general, the neighborhood of X ⊆ V G , denoted by N G (X), is defined to be v∈X N G (v), and the closed neighborhood of X, denoted by N G [X], is the set N G (X) ∪ 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 γ(G). A subset C ⊆ 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 β(G).
It is obvious that if G = ((A, B), E G ) is a bipartite graph, then γ(G) ≤ min{|A|, |B|}. In this paper the set of all bipartite graphs G = ((A, B), E G ) in which γ(G) = min{|A|, |B|} is denoted by B. Some properties of the graphs belonging to the set 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 B in terms of this new operation.

Bipartization of a graph
Let K H denote the set of all complete subgraphs of a graph H. If v ∈ V H , then the set

Definition 1. Let H be a graph and let
only if x is a vertex of the complete graph K (i = 1, . . . , f (K)).

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: is the bipartization of a graph H with respect to a function f : K H → N, then: . . , f (K)).   . 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. 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 ∪ B). Let B 1 , . . . , B l be the equivalence classes of this relation on B, 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 K H be the set of all complete subgraphs of H, and let f : 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 , . . . , B l , it , is an isomorphism between graphs G and B f (H).
We have proved that a bipartite graph G = ( (A, B) Consequently, every bipartite graph may be the bipartization of two non-isomorphic graphs.
Example 2. Fig. 2 depicts the bipartite graph G which is the bipartization of the nonisomorphic graphs H and F with respect to functions f : K H → N and g : K F → 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 : K H → 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 is an edge in the complete graph F i for i = 1, . . . , k. We now have the following two useful lemmas.
, and so they generate at least one cycle in B f (H).
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 S is the set of all blocks of a graph H, then
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 : K H → 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) ≤ 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) ≥ 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 , . . . , v l , v 0 ) be a shortest such cycle in B. Then l ≥ 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 0 v k is a chord of C. Now the choice of C implies that the vertices v 0 , v 1 , . . . , v k are mutually adjacent, and therefore, k = 2. Similarly, v 0 , v k , . . . , v l are mutually adjacent, and so we must have l = 3.    Fig. 2 shows the tree G which is the bipartization of two block graphs H and F with respect to functions f and g, respectively, which non-zero values are listed in the same figure.

Graphs belonging to the family 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 graph G belongs to the class B if and only if G is some bipartization of a graph. For that purpose, we need the following lemma.   We are ready to establish our main theorem that provides an alternative characterization of the graphs belonging to B in terms of the bipartization of a graph.  Assume now that H is a connected graph, and f : K H → 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 = K∈K H F K , is |A| ≤ |B| and γ(B f (H)) = |A|.
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 : K Km → {0, n}, where f (K) = 0 if and only if K ∈ K Km − {K m } (resp. g : K Kn → {0, m}, where g(K) = 0 if and only if K ∈ K Kn − {K n }). It is also evident that if min{m, n} ≥ 3, then K m,n does not belong to the family B (as γ(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.