The parity argument principle states that every finite graph has an even number of odd degree vertices. We consider the problem whose totality is guaranteed by the parity argument on a graph with potential. In this paper, we show that the problem of finding an unknown odd-degree vertex or a local optimum vertex on a graph with potential is polynomially equivalent to EndOfPotentialLine if the maximum degree is at most three. However, even if the maximum degree is 4, such a problem is $\mathtt{PPA}\cap \mathtt{PLS}$-complete. To show the complexity of this problem, we provide new results on multiple-source variants of EndOfPotentialLine, which is the canonical problem for EOPL. This result extends the work by Goldberg and Hollender; they studied similar variants of EndOfLine.

奇偶论证原理指出，每个有限图都有偶数个奇数度顶点。我们考虑一个有潜力的图，其奇偶性保证了总和的问题。在本文中，我们表明，如果最大度数最多为3，则在具有电势的图上找到未知奇数度顶点或局部最优顶点的问题在多项式上等同于EndOfPotentialLine。但是，即使最大程度为4，也会出现这样的问题$\mathtt{PPA}\cap \mathtt{最小二乘}$-完全的。为了显示这个问题的复杂性，我们提供多种来源的变种新成果EndOfPotentialLine，这是规范的问题EOPL。这一结果扩展了Goldberg和Hollender的工作；他们研究了EndOfLine的类似变体。