Realizability of Fault-Tolerant Graphs

Abstract

A connected graph \(G\) is optimal-\(\kappa \) if the connectivity \(\kappa (G)=\delta (G)\), where \(\delta (G)\) is the minimum degree of \(G\). It is super-\(\kappa \) if every minimum vertex cut isolates a vertex. An optimal-\(\kappa \) graph \(G\) is \(m\)-optimal-\(\kappa \) if for any vertex set \(S\subseteq V(G)\) with \(|S|\le m\), \(G-S\) is still optimal-\(\kappa \). The maximum integer of such \(m\), denoted by \(O_{\kappa }(G)\), is the vertex fault tolerance of \(G\) with respect to the property of optimal-\(\kappa \). The concept of vertex fault tolerance with respect to the property of super-\(\kappa \), denoted by \(S_{\kappa }(G)\), is defined in a similar way. In a previous paper, we have proved that \(\min \{\kappa _1(G)-\delta (G),\delta (G)-1\}\le O_\kappa (G)\le \delta (G)-1\) and \(\min \{\kappa _1(G)-\delta (G)-1,\delta (G)-1\}\le S_\kappa (G)\le \delta (G)-1\). We also have \(S_\kappa (G)\le O_\kappa (G)\le \delta (G)-1\). In this paper, we study the realizability problems concerning the above three bounds. By construction, we proved that for any non-negative integers \(a,b,c\) with \(a\le b\le c\), (i) there exists a graph \(G\) such that \(\kappa _1(G)-\delta (G)=a\), \(O_{\kappa }(G)=b\), and \(\delta (G)-1=c\); (ii) there exists a graph \(G\) with \(\kappa _1(G)-\delta (G)-1=a\), \(S_{\kappa }(G)=b\), and \(\delta (G)-1=c\); and (iii) there exists a graph \(G\) such that \(S_{\kappa }(G)=a\), \(O_{\kappa }(G)=b\), and \(\delta (G)-1=c\).