On the Complexity of Minimising the Moving Distance for Dispersing Objects

We study Geometric Graph Edit Distance (GGED), a graph-editing model to compute the minimum edit distance of intersection graphs that uses moving objects as an edit operation. We first show an $O(n\log n)$-time algorithm that minimises the total moving distance to disperse unit intervals. This algorithm is applied to render a given unit interval graph (i) edgeless, (ii) acyclic and (iii) $k$-clique-free. We next show that GGED becomes strongly NP-hard when rendering a weighted interval graph (i) edgeless, (ii) acyclic and (iii) $k$-clique-free. Lastly, we prove that minimising the maximum moving distance for rendering a unit disk graph edgeless is strongly NP-hard over the $L_1$ and $L_2$ distances.