A graph $G$ contains a graph $H$ as a pivot-minor if $H$ can be obtained from $G$ by applying a sequence of vertex deletions and edge pivots. Pivot-minors play an important role in the study of rank-width. Pivot-minors have mainly been studied from a structural perspective. In this paper we perform the first systematic computational complexity study of pivot-minors. We first prove that the Pivot-Minor problem, which asks if a given graph $G$ contains a pivot-minor isomorphic to a given graph $H$, is NP-complete. If $H$ is not part of the input, we denote the problem by $H$-Pivot-Minor. We give a certifying polynomial-time algorithm for $H$-Pivot-Minor when (1) $H$ is an induced subgraph of $P_3+tP_1$ for some integer $t\geq 0$, (2) $H=K_{1,t}$ for some integer $t\geq 1$, or (3) $|V(H)|\leq 4$ except when $H \in \{K_4,C_3+ P_1\}$. Let ${\cal F}_H$ be the set of induced-subgraph-minimal graphs that contain a pivot-minor isomorphic to $H$. To prove the above statement, we either show that there is an integer $c_H$ such that all graphs in ${\cal F}_H$ have at most $c_H$ vertices, or we determine ${\cal F}_H$ precisely, for each of the above cases.

翻译：图$G$包含图$H$作为主次图，如果$H$可以通过对$G$应用一系列顶点删除和边主变换得到。主次图在秩宽的研究中扮演重要角色，且以往主要从结构角度进行研究。本文首次对主次图进行系统性计算复杂性研究。我们首先证明主次判定问题（即给定图$G$是否包含与给定图$H$同构的主次图）是NP完全的。若$H$不包含在输入中，我们将该问题记为$H$-主次判定问题。我们针对以下情况给出了$H$-主次判定问题的可验证多项式时间算法：(1) $H$是$P_3+tP_1$（其中$t\geq 0$为整数）的诱导子图；(2) $H=K_{1,t}$（其中$t\geq 1$为整数）；(3) $|V(H)|\leq 4$且$H \notin \{K_4,C_3+ P_1\}$。设${\cal F}_H$为所有包含与$H$同构的主次图的诱导子图极小图集合。为证明上述结论，我们对上述每种情况或证明存在整数$c_H$使得${\cal F}_H$中所有图的顶点数不超过$c_H$，或精确确定了${\cal F}_H$的组成。