Since the problem of Konigsberg bridge was released in 1735, there have been many applications of graph theory in mathematics, physics, biology, computer science, and several fields of engineering. In particular, all communication networks can be modeled by graphs. The vulnerability is a concept that represents the reluctance of a network to disruptions in communication after a deterioration of some processors or communication links. Furthermore, the vulnerability values can be computed with many graph theoretical parameters. The rupture degree $r(G)$ of a graph $G=(V,E)$ is an important graph vulnerability parameter and defined as $r(G)=max\{\omega(G-S)-|S|-m(G-S):\omega(G-S)\geq2, S\subset V \}$, where $\omega(G-S)$ and $m(G-S)$ denote the number of connected components and the size of the largest connected component in the graph $G-S$, respectively. Recently, it has been proved that finding the rupture degree problem is $NP$-complete. In this paper, a heuristic algorithm to determine the rupture degree of a graph has been developed. Extensive computational experience on 88 randomly generated graphs ranging from 20\% to 90\% densities and from 100 to 200 vertices shows that the proposed algorithm is very effective.