Giyu Tomioka, the Water Hashira from the Demon Slayer series, has an iconic personality. He has displayed great feats of strength throughout the series. Introduced in the very first episode, Giyu's character has intrigued many fans, making him such popular.
One of Giyu's actions throughout the series has raised a question among fans that was unclear by the series. Why does Giyu Tomioka carry a rope with him? He might be carrying the rope to capture demons from the mountains, which are later used as a part of the Final Selection Exam.
Disclaimer: This article contains spoilers from the anime/manga. This article also contains speculative theory based on the author's opinions.
Giyu's probable reasoning for carrying a rope in Demon Slayer

During Episode 19 of the Demon Slayer: Mount Natagumo arc, Giyu took out a rope from his Kimono. It happened right after he saved Inosuke from the Father Spider Demon. As Inosuke stood there and challenged Giyu despite being fatally injured, Giyu took out the rope and tied Inosuke to a tree.
The main question that arises here is why Giyu randomly keeps a rope under his Kimono. An explanation can be concluded based on all the events presented in the series. Giyu has been passively tasked with capturing demons for the Demon Slayer Corps as part of the Final Examination.
As explained in the series, it was a Hashira's duty to bring low-level demons for the Final Selection Examination. The most educated guess can be made that he had the rope for capturing any low-level demons he might come across for the Corps. Not just Giyu, but probably all other Hashira carry a rope tucked away to capture demons.
As said earlier, the Hashira are responsible for bringing the demons to Mt. Fujikasane, where they are released. The Mountain is then used as part of the examination ground for the Final Selection of candidates.
Every Hashira is tasked with this duty, making it possible that every one of them carries a rope.
The role of a demon during the Final Selection

The premise of the Final Selection in Demon Slayer is to survive seven consecutive nights on Mt. Fujikasane. Those who survived were allowed to join the Demon Slayer Corps as certified Demon Slayers. However, as explained in the anime, the mountain is filled with demons captured by the Hashira.
Due to the Wisteria plant growing all around the mountain near the bottom, the demons are unable to escape. The demons that are spread on the mountain range range from all different levels of power. Mostly being low levels, there are occasional high-level demons as well. The Hand Demon seen during Tanjiro's trial was quite strong, as it had been known to kill Sabito.
The Hashira are not only in charge of bringing the demons but also of evaluating which demon should be suitable for a trial. As seen during Mt Natagumo arc, despite Giyu bringing his rope along with him, he chose to kill the Father Spider Demon rather than capturing it.
Final Thoughts

Despite the theory presented here, the rope's application was used for a comedic purpose. Many more such small implications from the series have been left unexplained by the mangaka. A deep dive into those instances can open up various possibilities that are both interesting and intriguing.
The Demon Slayer anime has a wider array of theories regarding all the major characters from the series. With the upcoming Infinity Castle arc around the corner, many more such implications can be anticipated.
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