The 2-tuple dominating independent number of a random graph

In this note, we show that 2-tuple dominating independent number of the Erd\H{o}s--R\'{e}nyi graph $G\left(n,p\right)$ a.a.s.~has a two-point concentration when $p$ is a constant

Independent coalition in graphs

An independent coalition in a graph $G=(V,E)$ consists of two disjoint sets of vertices $V_1$ and $V_2$, neither of which is an independent dominating set but whose union $V_1 \cup V_2$ is an independent dominating set. An independent coalition partition in a graph $G$ is a vertex partition $\pi= \lbrace V_1,V_2,\dots ,V_k \rbrace$ such that each set $V_i$ of $\pi$ either is an independent dominating set consisting of a single vertex of degree $n-1$, or is not an independent dominating set but forms an independent coalition with another set $V_j \in \pi$ which is not an independent dominating set. In this paper we study the concept of independent coalition partition (ic-partition). We introduce a family of graphs that have no ic-partition. We also determine the independent coalition number of some custom graphs and investigate graphs $G$ with $IC(G)\in\{1,2,3,4,n\}$ and the trees $T$ with $IC(T)=n-1$, where $n$ denotes the order of the graph.Comment: 17 page

The independent domination number of a random graph

We prove a two-point concentration for the independent domination number of the random graph $G_{n,p}$ provided p²ln(n) ≥ 64ln((lnn)/p)