Removing overlaps is a central task in domains such as scheduling, visibility, and map labelling. This can be modelled using graphs, where overlap removals correspond to enforcing a certain sparsity constraint on the graph structure. We continue the study of the problem Geometric Graph Edit Distance (GGED), where the aim is to minimise the total cost of editing a geometric intersection graph to obtain a graph contained in a specific graph class. For us, the edit operation is the movement of objects, and the cost is the movement distance. We present an algorithm for rendering the intersection graph of a set of unit circular arcs edgeless and $k$-clique-free in $O(n\log n)$ time, where $n$ is the number of arcs. The algorithm can be also used to solve an open case of the points-spreading problem on cyclic domains [Li \& Wang, CGT 2025]. We also show that GGED remains strongly NP-hard on unweighted interval graphs, solving an open problem of Honorato-Droguett et al. [WADS 2025]. We complement this result by showing that GGED is strongly NP-hard on sets of $d$-balls and $d$-cubes, for any $d\ge 2$. Finally, we present an XP algorithm (parameterised by the number of maximal cliques) that removes all edges from the intersection graph of a set of weighted unit intervals.

翻译：消除重叠是调度、可视性及地图标注等领域中的核心任务。这可通过图模型进行建模，其中消除重叠对应于对图结构施加特定的稀疏性约束。我们继续研究几何图编辑距离问题，其目标是通过最小化编辑几何相交图的总代价，使其包含在特定图类中。本研究中，编辑操作是移动对象，代价为移动距离。我们提出一种算法，可在$O(n\log n)$时间内将一组单位圆弧的相交图转化为无边且无$k$-团结构，其中$n$为圆弧数量。该算法还可用于解决循环域上点扩散问题的未解情形。此外，我们证明几何图编辑距离在无权区间图上仍为强NP困难问题，解决了Honorato-Droguett等人的开放问题。我们进一步补充证明：对于任意$d\ge 2$，几何图编辑距离在$d$-球和$d$-立方体集合上均为强NP困难问题。最后，我们提出一个参数化于最大团数量的XP算法，可移除一组加权单位区间相交图中的所有边。