By means of the \( q \)-finite differences and the derivative operator, we derive, from an alternating \( q \)-binomial sum identity with a free variable \( x \), several interesting identities concerning the generalized \( q \)-harmonic numbers.

In [A.G. Chetwynd and A.J.W. Hilton, Critical star multigraphs, Graphs and Combinatorics 2 (1986), 209-221] Chetwynd and Hilton started the investigations of the edge-chromatic properties of a particular class of multigraphs, which they called star multigraphs. A star multigraph is a multigraph such that there exists a vertex \(v^*\) that is incident with each multiple edge. Star multigraphs turn out to be useful tools in the study of the chromatic index of simple graphs.
The main goal of this paper is to provide shorter and simpler proofs of all the main theorems contained in the above mentioned paper. Most simplifications are achieved by means of a formula for the chromatic index recently obtained by the author and by a careful use of arguments involving fans.

Broadcasting is the process of message transmission in a communication network. The communication network is modeled by a graph \( G = (V, E) \), where the set of vertices \( V \) represents the network members and the set of edges \( E \) represents the communication links between two given vertices. We assume that \( G \) is connected and undirected. One vertex, called the \emph{originator} of the graph, holds a message that has to be transmitted to all vertices of the network by placing a series of calls over the network.

A \textbf{k-port} line broadcasting in \( G \) is a model in which an informed vertex can call, at each time unit, at most \( k \) vertices and transmit a message through a path, as long as two transmissions do not use the same edge at the same time. In case \( k \) is not bounded, the model is called the all-port line model.

In this paper, we extend Cohen’s work \([6]\), which handles the all-port line model.

In this paper we consider 1-movable dominating sets, motivated by the use of sensors employed to detect certain events in networks, where the sensors have a limited ability to react under changing conditions in the network. A 1-movable dominating set is a dominating set \( S \subseteq V(G) \) such that for every \( v \in S \), either \( S – \{v\} \) is a dominating set, or there exists a vertex \( u \in (V(G) – S) \cap N(v) \) such that \( (S – \{v\}) \cup \{u\} \) is a dominating set. We present computational complexity results and bounds on the size of 1-movable dominating sets in arbitrary graphs. We also give a polynomial time algorithm to find minimum 1-movable dominating sets for trees. We conclude by extending this idea to \( k \)-movable dominating sets.

A subset \(S\) of vertices in a graph \(G\) is called a \({geodetic\; dominating\; set}\) if \(S\) is both a geodetic set and a (standard) dominating set. In this paper, we study geodetic domination on graphs.

Let \(\Sigma\) be a totally ordered set. We work on finite strings \(b = b_1 b_2 \ldots b_m\) of \(b_i\) elements from \(\Sigma\). Such a \(b\) is a Lyndon word (Lyn) if \(m \geq 1\), and \(b\) is the unique first in lexicographic order among the \(m\) rows of the \(m \times m\) circulant matrix with \(b\) as the first row.A classic result is that every string \(b\) has a unique maximal factorization \(umf(b)\) into Lyndon words, each Lyndon word of the maximum possible size in \(b\).In 1983, J. P. Duval \([6]\) published Algorithm 1, which finds \(umf(b)\). It was studied in 1991 by A. Apostolico and M. Crochemore \([1]\). Their work was then studied in 1994 by J.W. Daykin, C.S. Iliopoulos, and W.F. Smyth \([5]\).Since Duval used a programming language, we start by giving a new simple account of his Algorithm 1. Our Algorithm 2 modifies Duval’s Algorithm 1 to find \(umf(a)\), when \(a\) is a string \(a = A_1 A_2 \ldots A_p\) of Lyndon words \(A_i\).Our Algorithm 3 is also for a string \(a = A_1 A_2 \ldots A_p\) of Lyndon words \(A_i\). It is completely different from Algorithms 1 and 2. It snakes right, left, right, and so on. It revealed that Lyndon words have a special structure. We give an example where Algorithm 3 needs almost \(2m\) tests; we think that is the most needed, but cannot give a rigorous proof.

Let \(\Sigma\) be a totally ordered set. We work on finite strings \(b = b_1 b_2 \ldots b_m\) of \(b_i\) elements from \(\Sigma\). Such a \(b\) is a lyn(Lyndon word) if \(m \geq 1\), and \(b\) is the unique first in lex(lexicographic order) among the \(m\) rows of the \(m \times m\) circulant matrix with \(b\) as first row.

A classic result is that every string \(b\) has a unique max factorization \(umf(b)\) into Lyns , each Lyn of maximum possible size in \(b\).

In 1983 J. P. Duval [6] published Algorithm 1, which finds \(umf(b)\). It was studied in 1991 by A. Apostolico and M. Crochemore [1]. Then their work was studied in 1994 by J.W. Daykin, C.S. Iliopoulos, and W.F. Smyth [5].

Since Duval used a programming language, we start by giving a new simple account of his Algorithm 1. Then our Algorithm 2 given here modifies Duval’s Algorithm 1 to find \(umf(a)\), when \(a\) is a string \(a = A_1 A_2 \ldots A_p\) of Lyndon words \(A_I\).

Our Algorithm 3 is also for a string \(a = A_1 A_2 \ldots A_p\) of Lyndon words \(A_I\). It is completely different from Algorithms 1 and 2. It snakes right, left, right, and so on. It revealed the fact that Lyndon words have a special structure. We give an example where Algorithm 3 needs almost \(2m\) tests; we think that is the most needed, but cannot give a rigorous proof.

We find interesting properties of Lyns, some of which may be new.

A family \(\mathcal{G}\) of connected graphs is a family with constant metric dimension if \(\dim(G)\) is finite and does not depend upon the choice of \(G\) in \(\mathcal{G}\).

The metric dimension of some classes of plane graphs has been determined in \([3]\), \([4]\), \([5]\), \([10]\), \([13]\), and \([18]\), while the metric dimension of some classes of convex polytopes has been determined in \([8]\), and a question was raised as an open problem: Is it the case that the graph of every convex polytope has constant metric dimension? In this paper, we study the metric dimension of two classes of convex polytopes. It is shown that these classes of convex polytopes have constant metric dimension and only three vertices chosen appropriately suffice to resolve all the vertices of these classes of convex polytopes. It is natural to ask for the characterization of classes of convex polytopes with constant metric dimension.

The degree set of a graph \( G \) is the set \( S \) consisting of the distinct degrees of vertices in \( G \). In 1977, Kapoor, Polimeni, and Wall \([2]\) determined the least number of vertices among simple graphs with a given degree set. In this note, we look at the analogue problem concerning the least order and the least size of a multigraph with a given degree set.

Let \(\mathcal{P}\) be a graph property and \(G\) a graph. \(G\) is said to be \(\mathcal{P}\)-saturated if \(G\) does not have property \(\mathcal{P}\) but the addition of any edge between non-adjacent vertices of \(G\) results in a graph with property \(\mathcal{P}\). If \(\mathcal{P}\) is a bipartite graph property and \(G\) is a bipartite graph not in \(\mathcal{P}\), but the addition of any edge between non-adjacent vertices in different parts results in a graph in \(\mathcal{P}\), then \(G\) is \(\mathcal{P}\)-bisaturated. We characterize all \(\mathcal{P}\)-saturated graphs, for which \(\mathcal{P}\) is the family of interval graphs, and show that this family is precisely the family of maximally non-chordal graphs. We also present a conjectured characterization of all \(\mathcal{P}\)-bisaturated graphs, in the case where \(\mathcal{P}\) is the family of interval bigraphs, and prove it as far as current forbidden subgraph characterizations allow. We demonstrate that extremal non-interval graphs and extremal non-interval bigraphs are highly related, in that the former is simply a complete graph with \(2K_2\) removed and the latter is a complete bipartite graph with \(3K_2\) removed.