Is it oriented graph or not?

Sorry, the graph should be directed. We are adding it now.

EDIT: Fixed. Again we are very sorry for those errors. We will make further efforts in preparing problems.
日本語の問題文を読まれている方へ: 無視してください（英語版の問題文に別の不備がありました）。

Can the weight of an edge be 0?

Sorry, the fourth condition should be: "Each edge has either an integer weight between 0 and 100 (inclusive), or a label X or Y." We are now fixing it.

EDIT: the statement is now fixed. We are sorry for the inconvenience.
日本語の問題文を読まれている方へ: 無視してください（英語版の問題文に不具合がありました）。