| Summary: | Social structure decomposition has been a valuable research topic in the study of social networks, and it still has many unresolved problems. At the same time, an increasing number of people pays attention to the security issue that is worth considering and researching. With security consideration, we study a new social structure decomposition problem that can be formulated as a minimization problem in graph theory as follows: given a social network with formulation as a graph <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula>, partition the vertex set of <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> into the minimum number of subsets, such that each subset induces an acyclic graph called a forest. This minimum number is called the vertex arboricity, denoted by <inline-formula> <tex-math notation="LaTeX">$va(G)$ </tex-math></inline-formula>. It is well known that the vertex arboricity <inline-formula> <tex-math notation="LaTeX">$va(G)\leq 3$ </tex-math></inline-formula> for each planar graph <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> in the graph theory. In this paper, we prove that for the planar graph <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula>, if no 3-cycle intersects a 4-cycle or no 3-cycle intersects a 5-cycle, then <inline-formula> <tex-math notation="LaTeX">$va(G)\leq 2$ </tex-math></inline-formula>.
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