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.

Forming all distinct subsets with \(k\) or fewer objects from a set with \(n\) elements can be accomplished by generating a subset of the binary reflected Gray code. This paper presents a straightforward algorithm that generates the desired Gray codewords by altering the stack which maintains the transition sequence that determines the next codeword position to be altered.

A vertex of a graph \(G\) dominates itself and its neighbors. A set \(S\) of vertices of \(G\) is a dominating set if each vertex of \(G\) is dominated by some vertex of \(S\). The domination number \(\gamma(G)\) of \(G\) is the minimum cardinality of a dominating set of \(G\). A minimum dominating set is one of cardinality \(\gamma(G)\). A subset \(T\) of a minimum dominating set \(S\) is a forcing subset for \(S\) if \(S\) is the unique minimum dominating set containing \(T\). The forcing domination number \(f(S, \gamma)\) of \(S\) is the minimum cardinality among the forcing subsets of \(S\), and the forcing domination number \(f(G, \gamma)\) of \(G\) is the minimum forcing domination number among the minimum dominating sets of \(G\). For every graph \(G\), \(f(G, \gamma) \leq \gamma(G)\).It is shown that for integers \(a, b\) with \(b\) positive and \(0 \leq a \leq b\), there exists a graph \(G\) such that \(f(G, \gamma) = a\) and \(\gamma(G) = b\). The forcing domination numbers of several classes of graphs are determined, including complete multipartite graphs, paths, cycles, ladders, and prisms. The forcing domination number of the cartesian product \(G\) of \(k\) copies of the cycle \(C_{2k+1}\) is studied. Viewing the graph \(G\) as a Cayley graph, we consider the algebraic aspects of minimum dominating sets in \(G\) and forcing subsets.

The complete stability \(cs(P_k)\), where \(P_k\) denotes the property of having a \(k\)-factor, satisfies \(cs(P_k) = n + k – 2, \text{ if } 1 \leq k \leq 3\), and \(n + k – 2 \leq cs(P_k) \leq n + k – 1, \text{ if } k \geq 4\). A similar result for bipartite graphs with complete biclosure is proved also.

It is known that in every 2-coloring of the edges of the complete graph there exist two vertex disjoint paths—one red, one blue—that collectively cover all the vertices. In this paper, analogous existence and efficiency questions are examined when edges are missing from the complete graph. The main result shows the existence of a path cover when at most \(\left\lfloor{n}/{2}\right\rfloor\) edges are missing. An example shows this result is sharp. A second result shows that a path cover can be found efficiently if a matching is missing.

A map shows only the names of some (but not all) towns in a region. If for each town, the names of all neighboring towns are known, when is it possible to reconstruct from this information the missing names? We deal with a generalization of this problem to arbitrary graphs. For a graph \(G = (V, E)\) with \(n\) nodes, we give an \(O(n^3)\) algorithm to recognize those instances for which the answer is YES, as well as two characterization theorems: NO-instances are determined by the existence of a certain partition of \(V\) and YES-instances by the existence of a suitable weak order in \(V\).

Let \(G\) be a connected claw-free graph of order \(n\). If \(G \not\in M\) and the minimum degree of \(G\) is at least \(\frac{n}{4}\), then \(G\) is traceable.Here, \(M\) is a set of graphs such that each element in \(M\) can be decomposed into three disjoint subgraphs \(G_1\), \(G_2\), \(G_3\) and \(E_G(G_i, G_j) = u_iu_j\), here \(1 \leq i, j \leq 3\) and \(u_i \in G_i\), \(1\leq i \leq 3\).

In this paper, we consider the two-dimensional sequence of primitive polynomials, which is defined by two positive integers and a primitive polynomial. The concept of \(q^m\) conjugate order is used to describe the two-dimensional sequence. Using the two-dimensional sequences, we can find maximum period primitive-polynomial sequences for more values of degrees than using the one-dimensional sequences. Examples of the applications of the two-dimensional sequence by computer search are shown.