A proper vertex colouring is called a 2-dynamic colouring, if for every vertex v with degree at least 2, the neighbours of v receive at least two colours. The smallest integer k such that G has a dynamic colouring with k colours denoted by χ 2 (G). We denote the cartesian product of G and H by G H. In this paper, we find the 2-dynamic chromatic number of cartesian product of complete graph with complete graph K r K s , complete graph with complete bipartite graph K n K 1,s and wheel graph with complete graph W l K n . ARTICLE HISTORY

Let \(G=(V,E)\) be a simple graph. A set \(S\subseteq V\) is a dominating set if every vertex in \(V \setminus S\) is adjacent to a vertex in \(S\). The domination number of a graph \(G\), denoted by \(\gamma(G)\) is the minimum cardinality of a dominating set of \(G\). A set \(D \subseteq E\) is an edge dominating set if every edge in \(E\setminus D\) is adjacent to an edge in \(D\). The edge domination number of a graph \(G\), denoted by \(\gamma'(G)\) is the minimum cardinality of an edge dominating set of \(G\). We characterize trees with  domination number equal to twice edge domination number.

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An edge-vertex Roman dominating function (or just ev-RDF) of a graph [Formula: see text] is a function [Formula: see text] such that for each vertex [Formula: see text] either [Formula: see text] where [Formula: see text] is incident with [Formula: see text] or there exists an edge [Formula: see text] adjacent to [Formula: see text] such that [Formula: see text]. The weight of a ev-RDF is the sum of its function values over all edges. The edge-vertex Roman domination number of a graph [Formula: see text], denoted by [Formula: see text], is the minimum weight of an ev-RDF [Formula: see text]. We provide a characterization of all trees with [Formula: see text], where [Formula: see text] is the domination number of [Formula: see text]

A vertex u of a graph G = ( V,E ), ve -dominates every edge incident to u , as well as every edge adjacent to these incident edges. A set S ⊆ V is a vertex-edge dominating set (or a ved–set for short) if every edge of E is ve- dominated by at least one vertex of S . The vertex-edge domination number is the minimum cardinality of a ved–set in G. In this paper, we investigate the graphs having unique minimum ved-sets that we will call UVED-graphs. We start by giving some basic properties of UVED-graphs. For the class of trees, we establish two equivalent conditions characterizing UVED-trees which we subsequently complete by providing a constructive characterization.