Graphs with equal domination and covering numbers

Abstract

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.

Bipartite graphs with close domination and k-domination numbers

“…Note that in [16], where ( ) γ γ , 2 -graphs were studied, a graph class was introduced that exactly corresponds to the class 2 defined here. Moreover, by the results in [17,20,23], the class 2 is the collection of those connected graphs which satisfy ( ) = ( ) τ G γ G and ( ) ≥ δ G 2. Hansberg proved the following lemma which directly implies that the class k contains all…”

Bipartization of Graphs

“…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.…”