How to build a graph with these properties? Show that if $I$, $m$ and $n$ are integers such that $0 < I \leq m \leq n$, then there exists a simple graph $G$ with $\kappa(G) = I$ , $\kappa'(G) = m$, and $\delta(G) = n$.
$\kappa(G)$ is the minimum number of vertices that must be omitted from the graph to make it disconnedted and $\kappa’(G)$ is is the minimum number of edges that must be omitted from the graph to make it disconnedted .
And $\delta(G)$ is the minimum degree of vertices in a graph.
This problem is problem 3.1.7 from Bondy-Murty book that I tryed to solve but I couldn’t.
 A: If $\kappa'=1$, take two disjoint copies of $K_{\delta+2}$ (the complete graph on $\delta+2$ vertices), remove an edge $u_1v_1$ in the first one, an edge $x_1y_1$ in the second one, then add edge $u_1x_1$. Removing this edge or vertex $u_1$ or $x_1$ disconnects the graph and each vertex has degree $\delta+1$ except vertices $v_1$ and $y_1$ where the degree is $\delta$. If $\delta=1$ this construction results in $P_6$ which shows us that the construction is not optimal, because $P_3$ would suffice.
If $\kappa=1$, add another $\kappa'-1$ disjoint copies of the previously constructed graph and add edges $u_ix_j$ for all $i\neq j$, $1\leq i,j\leq \kappa'$. Then we need to remove $\kappa'$ edges at $u_i$ or $x_i$ or any of these vertices to disconnect the graph.
The construction gets trickier for $\kappa>1$. First let's assume $\kappa=\kappa'$. Take again two disjoint copies of $K_{\delta+2}$. Select a (near-)perfect matching in each of these complete graphs. Since both complete graphs are of the same size, we have a bijection between the edges of the matchings. Select $\kappa$ if $\delta$ is even or $\kappa-1$ if $\delta$ is odd of these edgepairs. We change the adjacencies of each edge pair like before, i.e. if we have an edge pair ($uv$,$xy$) we remove these edges and add $ux$, resulting in a(n unchanged) degree of $\delta+1$ for $u$ and $x$ and a degree of $\delta$ for $v$ and $y$. If $\delta$ is odd, add an edge between the two unmatched vertices in each copy. This results in the graph with the desired properties.
If $\kappa < \kappa'$, we extend our construction similar as we did before. Take another $\kappa'-\kappa$ disjoint copies of the graph we just constructed. In each one and the original one, select one of the $\kappa$ edges we constructed in the previous step, which gives us $n:=\kappa'-\kappa+1$ edges $u_ix_i$. Like before, add all edges $u_ix_j$ with $i\neq j$ and $1\leq i,j\leq n$, which leads to the graph with the desired properties.
The idea of using perfect matchings in complete graphs I got from Akiyama in "Regular graphs containing a given graph" (1983).
A: The following graph works:
Select two $K_{n+1}$ graphs and name one of them $G_1$ and name the other one $G_2$ . Select $l$ arbitrary vertices in $G_1$ and name them $v_1,\dots,v_l$ and select $m$ arbitrary vertices in $G_2$ and name them $u_1,\dots,u_m$ .
For each $1\leq i\leq l$ connect $v_i$ to $u_i$ . Then connect $v_l$ to $u_{l+1},\dots, u_m$.
Now this graph has $2n+2$ vertices such that each has degree greater than or equal to $n$. And it can be easily proved that for this graph we have : $\kappa=l$ and $\kappa’=m$.
