Instant-runoff voting (IRV) is often used when voters rank candidates rather than choosing only one favourite. We study IRV under graph-induced metric preferences where each vertex of an unweighted undirected graph hosts one voter and is also a possible candidate location. Voters rank candidates by shortest-path distance with fixed deterministic tie-breaking. We focus on exclusion zones, i.e., sets S such that, whenever at least one candidate lies in S, the IRV winner must also lie in S. Such zones serve as robustness certificates, identifying regions whose participation prevents outside winners from emerging. For general graphs, exclusion-zone verification is co-NP-complete and minimum-zone computation is NP-hard. We show that both problems become polynomial-time solvable on trees. Our main tool is a membership test asking whether a candidate can be forced to lose using opponents from a restricted region. A round-1 reduction shows that any such loss has a witness in which the candidate is eliminated in the first IRV round, enabling a bottom-up dynamic program on trees. We also show that minimum-zone computation has a much smaller search space than its definition suggests. The pairwise-loss graph, obtained from all two-candidate elections, imposes closure constraints on every exclusion zone. With deterministic tie-breaking this graph is a tournament, implying that every nonempty exclusion zone on a tree is generated by the closure of one vertex. Thus, the minimum exclusion zone can be found by testing only linearly many candidate sets. On the opposite front, we refine the intractability range of computing minimum exclusion zones on general graphs, extending it to a much broader class of deterministic elimination rules, dubbed as Strong Forced Elimination.

翻译：即时决选投票（IRV）常用于选民对候选人进行排序而非仅选一名最喜爱者的情况。我们研究图诱导度量偏好下的IRV：在无权重无向图中，每个顶点容纳一位选民，同时也是可能的候选位置。选民基于最短路径距离（采用固定确定性平局处置）对候选人排序。我们聚焦于排斥区域，即满足以下条件的集合S：只要至少有一个候选者位于S中，IRV胜出者必然也在S中。此类区域可作为鲁棒性证书，识别出那些能阻止外部胜出者出现的参与区域。对一般图而言，排斥区域验证是co-NP完全的，最小区域计算是NP难的。我们证明这两个问题在树上均可在多项式时间内求解。核心工具是一个成员资格测试：判断某个候选者能否被限制区域内的对手强制淘汰。通过第一轮约简化，我们发现任何此类淘汰都存在一个证明该候选者在IRV首轮即被淘汰的见证，从而支持在树上进行自底向上的动态规划。我们还证明最小区域计算的搜索空间远小于其定义所示。由所有两候选者选举生成的成对损失图，对每个排斥区域施加闭包约束。在确定性平局处置下，该图构成一个锦标赛图，这意味着树上每个非空排斥区域均由单个顶点的闭包生成。因此，最小排斥区域仅需测试线性数量候选集即可找到。另一方面，我们深化了一般图上最小排斥区域计算的理论难度范围，将其扩展至更广泛的确定性淘汰规则类别（称为强强制淘汰）。