A covering of the complete directed symmetric graph \(DK_v\) by \(m\)-circuits, denoted by \((v,m) – {DCC}\), is a family of \(m\)-circuits in \(DK_v\) whose union is \(DK_v\). The covering number \(C(v,m)\) is the minimum number of \(m\)-circuits in such a covering.The covering problem is to determine the value \(C(v,m)\) for every integer \(v \geq m\). In this paper, the problem is reduced to the case \(m+5 \leq v \leq 2m – 4\), for any fixed even integer \(m \geq 4\).In particular, the values of \(C(v,m)\) are completely determined for \(m = 12, 14,\) and \(16\). Additionally, a directed construction of optimal \((6k + 11, 4k + 6) – {DCC}\) is given.

It is shown that if \(H\) is a connected graph obtained from \(H_1\) and \(H_2\) by joining them with a bridge, then \(r(K_k, H) \leq r(K_k, H_1) + r(K_k, H_2) + k – 2\). We give some applications of this result.

Two closely related types of vertex subsets of a graph, namely external redundant sets and weak external redundant sets, together with associated parameters, are discussed. Both types may be used to characterize those irredundant subsets of a graph which are maximal.

Let \(G\) be a graph and let \(S\) be a subset of vertices of \(G\). A vertex \(v\) of \(G\) is called perfect with respect to \(S\) if \(|N[v] \cap S| = 1\), where \(N[v]\) denotes the closed neighborhood of \(v\). The set \(S\) is defined to be a perfect neighborhood set of \(G\) if every vertex of \(G\) is perfect or adjacent with a perfect vertex.The perfect neighborhood number \(\theta(G)\) of \(G\) is defined to be the minimum cardinality among all perfect neighborhood sets of \(G\). In this paper, we present a variety of algorithmic results on the complexity of perfect neighborhood sets in graphs.

We consider the problem of sweeping (or grazing) the interior/exterior of a polygon by a flexible rope whose one endpoint (anchor) is attached on the boundary of the polygon. We present a linear-time algorithm to compute the grazing area inside a simple polygon. We show how to extend the algorithm for computing the internal grazing area, without increasing its time complexity, to compute the grazing area in the exterior of a simple polygon. For grazing in the exterior of a convex polygon, we present an \(O(n)\) time algorithm to locate the anchor point that maximizes the simple grazing area. All three algorithms are optimal within a constant factor. Grazing area problems can be viewed as guard placement problems under \(L\)-visibility.