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A dominating set of a graph G is a set D ⊆ V G such that every vertex in V G − D is adjacent to at least one vertex in D, and the domination number γ(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.

A dominating set of a graph G is a set D ⊆ V G such that every vertex in V G − D is adjacent to at least one vertex in D, and the domination number γ (G) of G is the minimum cardinality of a dominating set of G. A set C ⊆ V G is a covering set of G if every edge of G has at least one vertex in C. The covering number β(G) of G is the minimum cardinality of a covering set of G. The set of connected graphs G for which γ (G) = β(G) is denoted by C γ =β , whereas B denotes the set of all connected bipartite graphs in which the domination number is equal to the cardinality of the smaller partite set. In this paper, we provide alternative characterizations of graphs belonging to C γ =β and B. Next, we present a quadratic time algorithm for recognizing bipartite graphs belonging to B, and, as a side result, we conclude that the algorithm of Arumugam et al. (Discrete Appl Math 161:1859-1867, 2013) allows to recognize all the graphs belonging to the set C γ =β in quadratic time either. Finally, we consider the related problem of patrolling grids with mobile guards, and show that it can be solved in O(n log n + m) time, where n is the number of line segments of the input grid and m is the number of its intersection points.

A set D of vertices of a graph G is a dominating set of G if every vertex in V G − D is adjacent to at least one vertex in D. The domination number (upper domination number, respectively) of a graph G, denoted by γ(G) (Γ(G), respectively), is the cardinality of a smallest (largest minimal, respectively) dominating set of G. A subset D ⊆ V G is called a certified dominating set of G if D is a dominating set of G and every vertex in D has either zero or at least two neighbors in V G − D. The cardinality of a smallest (largest minimal, respectively) certified dominating set of G is called the certified (upper certified, respectively) domination number of G and is denoted by γ cer (G) (Γ cer (G), respectively). In this paper relations between domination, upper domination, certified domination and upper certified domination numbers of a graph are studied. Closing open problemsWe close with the following list of open problems that we have yet to settle. Problem 21. Determine the class of graphs G for which γ cer (G) = Γ cer (G). Problem 22. Determine all the trees T for which γ cer (T ) = γ(T ). Problem 23. Let a, b, c, d be positive integers with a ≤ b ≤ c ≤ d. Find necessary and sufficient conditions on a, b, c, d such that there exists a graph G with γ(G) = a, Γ(G) = b, γ cer (G) = c and Γ cer (G) = d. Similarly, find necessary and sufficient conditions on a, b, c, d such that there exists a graph G with γ(G) = a, γ cer (G) = b, Γ(G) = c and Γ cer (G) = d. Finally, find necessary and sufficient conditions on a, b, c, d such that there exists a graph G with γ(G) = a, γ cer (G) = b, Γ cer (G) = c and Γ(G) = d.

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