a set $s$ of vertices in a graph $g$ is a dominating set if every vertex of $v-s$ is adjacent to some vertex in $s$. the domination number $gamma(g)$ is the minimum cardinality of a dominating set in $g$. the annihilation number $a(g)$ is the largest integer $k$ such that the sum of the first $k$ terms of the non-decreasing degree sequence of $g$ is at most the number of edges in $g$. in this p...

A edge 2-rainbow dominating function (E2RDF) of a graph G is a ‎function f from the edge set E(G) to the set of all subsets‎ ‎of the set {1,2} such that for any edge.......................

a {em 2-rainbow dominating function} (2rdf) of a graph $g$ is a function $f$ from the vertex set $v(g)$ to the set of all subsets of the set ${1,2}$ such that for any vertex $vin v(g)$ with $f(v)=emptyset$ the condition $bigcup_{uin n(v)}f(u)={1,2}$ is fulfilled, where $n(v)$ is the open neighborhood of $v$. the {em weight} of a 2rdf $f$ is the value $omega(f)=sum_{vin v}|f (v)|$. the {em $2$-r...

In this note, we introduce a graph invariant called the annihilation number and show that it is a sharp upper bound on the independence number. While the invariant does not distinguish between different graphs with the same degree sequence – since it is determined solely by the degrees – it still outperforms many well known upper bounds on independence number. The process which leads to the ann...