An exponential dominating set of graph [Formula: see text] is a kind of distance domination subset [Formula: see text] such that [Formula: see text], [Formula: see text], where [Formula: see text] is the length of a shortest path in [Formula: see text] if such a path exists, and [Formula: see text] otherwise. The minimum exponential domination number, [Formula: see text] is the smallest cardinality of an exponential dominating set. The minimum exponential domination number, [Formula: see text] can be decreased or increased by removal of some vertices from [Formula: see text]. In this paper, we investigate of this phenomenon which is referred to critical and stability in graphs.

No abstract

The integrity $I(G)$ of a noncomplete connected graph $G$ is a measure of network vulnerability and is defined by $I(G)=\min\limits_{S\subset V(G)}\{ |S|+m(G-S)\}$, where $S$ and $m(G-S)$ denote the subset of $V$ and the order of the largest component of $G-S$, respectively. The vertex neigbor integrity denoted as $VNI(G)$ is the concept of the integrity of a connected graph $G$ and is defined by $VNI(G)=\min\limits_{S\subset V(G)}\{|S|+m(G-S)\}$, where $S$ is any vertex subversion strategy of $G$ and $m(G-S)$ is the number of vertices in the largest component of $G-S$. If a network is modelled as a graph, then the integrity number shows not only the difficulty to break down the network but also the damage that has been caused. This article includes several results on the integrity of the $k-ary $ $tree$ $H_{n}^{k}$, the diamond-necklace $N_{k}$, the diamond-chain $L_{k}$ and the thorn graph of the cycle graph and the vertex neighbor integrity of the $H_{n}^{2}$, $H_{n}^{3}$.

A dominating set of a graph $G$ which intersects every independent set of a maximum cardinality in $G$ is called an independent transversal dominating set. The minimum cardinality of an independent transversal dominating set is called the independent transversal domination number of $G$ and is denoted by $\gamma_{it}(G)$. In this paper we investigate the independent transversal domination number for the transformation graph of the path graph $P_{n}^{+-+}$, the cycle graph $C_{n}^{+-+}$, the star graph $S_{1,n}^{+-+}$, the wheel graph $W_{1,n}^{+-+}$ and the complete graph $K_{n}^{+-+}$.