in this paper, some categorical properties of the category ${bf cpo}_{{bf act}text{-}s}$ of all {cpo $s$-acts}, cpo's equipped with actions of a monoid $s$ on them, and strict continuous action-preserving maps between them is considered. in particular, we describe products and coproducts in this category, and consider monomorphisms and epimorphisms. also, we show that the forgetful functor from...

We study the problem of adding an inclusion minimal set of edges to a given arbitrary graph so that the resulting graph is a split graph, called a minimal split completion of the input graph. Minimal completions of arbitrary graphs into chordal and interval graphs have been studied previously, and new results have been added recently. We extend these previous results to split graphs by giving a...

let g be a (p, q) graph. let f : v (g) → {1, 2, . . . , k} be a map. for each edge uv, assign the label gcd (f(u), f(v)). f is called k-prime cordial labeling of g if |vf (i) − vf (j)| ≤ 1, i, j ∈ {1, 2, . . . , k} and |ef (0) − ef (1)| ≤ 1 where vf (x) denotes the number of vertices labeled with x, ef (1) and ef (0) respectively denote the number of edges labeled with 1 and not labeled with 1....

This paper concerns finite, edge-transitive direct and strong products, as well as infinite weak Cartesian products. We prove that the direct product of two connected, non-bipartite graphs is edge-transitive if and only if both factors are edgetransitive and at least one is arc-transitive, or one factor is edge-transitive and the other is a complete graph with loops at each vertex. Also, a stro...

For fixed integers r, ` ≥ 0, a graph G is called an (r, `)-graph if the vertex set V (G) can be partitioned into r independent sets and ` cliques. This brings us to the following natural parameterized questions: Vertex (r, `)-Partization and Edge (r, `)-Partization. An input to these problems consist of a graph G and a positive integer k and the objective is to decide whether there exists a set...

We prove that a graph G contains no induced four-edge path and no induced complement of a four-edge path if and only if G is obtained from five-cycles and split graphs by repeatedly applying the following operations: substitution, split graph unification, and split graph unification in the complement (“split graph unification” is a new class-preserving operation that is introduced in this paper).

For a graph G, let D(G) be the set of all strong orientations of G. The orientation number of G, denoted by d(G), is defined as min{d(D) | D ∈ D(G)}, where d(D) denotes the diameter of the digraph D. In this paper, a sufficient condition is given for d(G×Kn) = d(G×Kn), where G is a graph with d(G) two or three and × is the tensor product of graphs.

Let f : V → {1, . . . , k} be a labeling of the vertices of a graph G = (V,E) and denote with f(N(v)) the sum of the labels of all vertices adjacent to v. The least value k for which a graph G admits a labeling satisfying f(N(u)) 6= f(N(v)) for all (u, v) ∈ E is called additive chromatic number of G and denoted η(G). It was first presented by Czerwiński, Grytczuk and Zelazny who also proposed a...

mathematical chemistry is a branch of theoretical chemistry for discussion and prediction of the molecular structure using mathematical methods without necessarily referring to quantum mechanics. in theoretical chemistry, distance-based molecular structure descriptors are used for modeling physical, pharmacologic, biological and other properties of chemical compounds. the wiener polarity index ...

A set D ⊆ V of a graph G = (V,E) is called a restrained dominating set of G if every vertex not in D is adjacent to a vertex in D and to a vertex in V \D. The MINIMUM RESTRAINED DOMINATION problem is to find a restrained dominating set of minimum cardinality. Given a graph G, and a positive integer k, the RESTRAINED DOMINATION DECISION problem is to decide whether G has a restrained dominating ...