A dominating set of a graph \( G \) is a set of vertices \( D \) such that for all \( \nu \in V(G) \), either \( \nu \in D \) or \( [\nu,d] \in E(G) \) for some \( d \in D \). The cardinality-redundance of a vertex set \( S \), \( CR(S) \), is the number of vertices \( x \in V(G) \) such that \( |N[x] \cap S| \geq 2 \). The cardinality-redundance of \( G \) is the minimum of \( CR(S) \) taken over all dominating sets \( S \). A set of vertices \( S \) such that \( CR(S) = CR(G) \) is a \( \gamma_{CR} \)-set, and the size of a minimum \( \gamma_{CR} \)-set is denoted \( \gamma_{CR}(G) \).

Here, we are concerned with extremal problems concerning cardinality-redundance. We give the maximum number of edges in a graph with a given number of vertices and given cardinality-redundance. In the cases that \( CR(G) = 0 \) or \( 1 \), we give the minimum and maximum number of edges of graphs when \( \gamma_{CR}(G) \) is fixed, and when \( CR(G) = 2 \), we give the maximum number of edges of graphs where \( \gamma_{CR}(G) \) is fixed. We give the minimum and maximum values of \( \gamma_{CR}(G) \) when the number of edges are fixed and \( CR(G) = 0, 1 \), and we give the maximum values of \( \gamma_{CR}(G) \) when the number of edges are fixed and \( CR(G) = 2 \).

It is known that for a maximal planar graph \( G \) with order \( n \geq 4 \), the independence number satisfies \( \frac{n}{4} \leq \alpha(G) \leq \frac{2n-4}{3} \). We show that the lower bound is sharp and characterize the extremal graphs for \( n \leq 12 \). For the upper bound, we characterize the extremal graphs of all orders.

The independence number \( \alpha(G) \) of a graph \( G \) is the size of the largest independent set. This parameter is difficult to determine in general, but can be bounded on various graph classes. This paper considers planar and maximal planar graphs.

Constraint Programming (CP) is a method used to model and solve complex combinatorial problems. An alternative to Integer Programming for solving large-scale industrial problems, it is, under some circumstances, more efficient than IP, but its strength lies mainly in the use of predicates to model problems. This paper presents the at-least-\( m \)-different predicate and provides a class of facet-defining inequalities of the convex hull of integer solutions. This predicate bounds the number of values that variables in a set may receive. The paper also presents a polynomial-time separation algorithm to be used in the context of a branch-and-bound optimization approach.

Let \( N_2DL(v) \) denote the set of degrees of vertices at distance \( 2 \) from \( v \). The \( 2 \)-neighborhood degree list of a graph is a listing of \( N_2DL(v) \) for every vertex \( v \). A degree restricted \( 2 \)-switch on edges \( v_1v_2 \) and \( w_1w_2 \), where \( \deg(v_1) = \deg(w_1) \) and \( \deg(v_2) = \deg(w_2) \), is the replacement of a pair of edges \( v_1v_2 \) and \( w_1w_2 \) by the edges \( v_1w_2 \) and \( v_2w_1 \), given that \( v_1w_2 \) and \( v_2w_1 \) did not appear in the graph originally. Let \( G \) and \( H \) be two graphs of diameter \( 2 \) on the same vertex set. We prove that \( G \) and \( H \) have the same \( 2 \)-neighborhood degree list if and only if \( G \) can be transformed into \( H \) by a sequence of degree restricted \( 2 \)-switches.

Let \( \Gamma \) be a finite group and let \( \Delta \) be a generating set for \( \Gamma \). A Cayley map is an orientable 2-cell imbedding of the Cayley graph \( G_\Delta(\Gamma) \) such that the rotation of arcs emanating from each vertex is determined by a unique cyclic permutation of generators and their inverses. A probability model for the set of all Cayley maps for a fixed group and generating set, where the distribution is uniform. We focus on certain finite abelian groups with generating set chosen as the standard basis. A lower bound is provided for the probability that a Cayley map for such a group and generating set is symmetrical.

A graph is asymmetric if the automorphism group of its set of vertices is trivial. A graph is called non-asymmetric if and only if it is not asymmetric. A graph \( G \) is minimally non-asymmetric if \( G \) is non-asymmetric but \( G – e \) is asymmetric for any edge \( e \) contained in \( G \).

Given a finite set \( V \) (of elements called varieties) and integers \( k \), \( r \), and \( \lambda \), a balanced incomplete block design (BIBD) is a family of \( k \)-element subsets of \( V \), called blocks, such that any element is contained in \( r \) blocks and any pair of distinct varieties \( u \) and \( w \) is contained in exactly \( \lambda \) blocks.

In this paper, we give examples of minimally non-asymmetric graphs constructed from balanced incomplete block designs.

There is a special case of a generalized Clifford algebra, known as a Clifford graph algebra, which is useful for studying a simple graph \( G_n \), with \( n \) vertices. We will discuss how this algebra \( GA(G_n) \) can represent \( G_n \), and prove that it exists in general by defining it as an appropriate sub-algebra of a classical Clifford algebra. We will then refine this process of “construction by inclusion” for the path graph \( P_n \), and the complete star graph \( K_{1,n} \), by choosing from a parent classical Clifford algebra as many bi-vectors as possible for the generators which define \( GA(P_n) \) and \( GA(K_{1,n}) \).

Elements of the Riordan group \(\mathcal{R}\) over a field \(\mathbb{F}\) of characteristic zero are infinite lower triangular matrices which are defined in terms of pairs of formal power series. We wish to bring to the forefront, as a tool in the theory of Riordan groups, the use of multiplicative roots \(a(x)^{\frac{1}{n}}\) of elements \(a(x)\) in the ring of formal power series over \(\mathbb{F}\). Using roots, we give a Normal Form for non-constant formal power series, we prove a surprisingly simple Composition-Cancellation Theorem and apply this to show that, for a major class of Riordan elements (i.e., for non-constant \(g(x)\) and appropriate \(F(x)\)), only one of the two basic conditions for checking that \((g(x), F(x))\) has order \(n\) in the group \(\mathcal{R}\) actually needs to be checked. Using all this, our main result is to generalize C. Marshall [6] and prove: Given non-constant \(g(x)\) satisfying necessary conditions, there exists a unique \(F(x)\), given by an explicit formula, such that \((g(x), F(x))\) is an involution in \(\mathcal{R}\). Finally, as examples, we apply this theorem to “aerated” series \(h(x) = g(x^r)\) to find the unique \(K(x)\) such that \((h(x), K(x))\) is an involution.