Many real-world interconnections among entities can be characterized as graphs. Collecting local graph information with balanced privacy and data utility has garnered notable interest recently. This paper delves into the problem of identifying and protecting critical information of entity connections for individual participants in a graph based on cohesive subgraph searches. This problem has not been addressed in the literature. To address the problem, we propose to extract the critical connections of a queried vertex using a fortress-like cohesive subgraph model known as $p$-cohesion. A user's connections within a fortress are obfuscated when being released, to protect critical information about the user. Novel merit and penalty score functions are designed to measure each participant's critical connections in the minimal $p$-cohesion, facilitating effective identification of the connections. We further propose to preserve the privacy of a vertex enquired by only protecting its critical connections when responding to queries raised by data collectors. We prove that, under the decentralized differential privacy (DDP) mechanism, one's response satisfies $(\varepsilon, δ)$-DDP when its critical connections are protected while the rest remains unperturbed. The effectiveness of our proposed method is demonstrated through extensive experiments on real-life graph datasets.