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 $γ(G)$ of $G$ is the minimum cardinality of a dominating set of $G$. A set $C\subseteq V_G$ 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 ${\cal C}_{γ=β}$, while ${\cal B}$ 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 ${\cal C}_{γ=β}$ and ${\cal B}$. Next, we present a quadratic time algorithm for recognizing bipartite graphs belonging to ${\cal B}$, and, as a side result, we conclude that the algorithm of Arumugam et al. [2] allows to recognize all the graphs belonging to the set ${\cal C}_{γ=β}$ in quadratic time either. Finally, we consider the related problem of patrolling grids with mobile guards, and show that this problem 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.