Optimal Graphs with Prescribed Connectivities

By the vertex connectivity $k(G)$ of a graph $G$ we mean the smallest number of vertices whose removal results in a disconnected or trivial graph, i.e., to a graph with one vertex. By the edge connectivity $\lambda(G)$ of a nontrivial graph $G$ we mean the smallest number of edges whose removal results in a disconnected graph. The vertex connectivity $k(G)$, the edge connectivity $\lambda(G)$, and the minimum vertex degree $\delta(G)$ of a graph are related by the following inequality found by Whitney in 1932: $k(G) \leq \lambda(G) \leq \delta(G)$. More recently, Chartrand and Harary proved that for any positive integers $a$, $b$, and $c$ such that $0 < a \leq b \leq c$ there exists a graph $G$ such that $k(G)=a$, $\lambda(G)=b$, and $\delta(G)=c$. In the proof, one constructs a family of such graphs with $2(c+1)$ vertices and $c(c+1)+b$ edges each. However, it is easily seen that one can sometimes construct a suitable graph with fewer vertices and edges. For example, if $a=b=c$, then the complete graph with $c+ 1$ vertices is optimal. In the present paper, we consider the problem of minimizing the Chartrand–Harary construction. We show that for $b=c$, it is possible to construct a graph with fewer vertices. Four cases are considered separately: $a=b=c$, $a<b<c$, $a=b<c$, and $a<b=c$. For each case, we present graphs with the prescribed characteristics $k(G)=a$, $\lambda(G)=b$, and $\delta(G)=c$ and with minimum numbers of vertices and edges.