Nicolás Honorato-Droguett

Nicolás Honorato-Droguett, Kazuhiro Kurita, Tesshu Hanaka et al.

In well-studied graph modification problems, adding and deleting vertices and edges are used as graph editing operations. We propose a model for graph modification on geometric intersection graphs called Geometric Graph Edit Distance that moves objects as an edit operation. Our results are mainly focused on interval graphs. In particular, we give a linear-time algorithm to find the minimum total moving distance to render an interval graph complete. The approach of this algorithm can be applied for: (i) rendering a unit square graph complete over the $L_1$ distance and (ii) attaining the existence of a $k$-clique on unit interval graphs. In addition, we provide LP-formulations to achieve several properties in the associated graph of unit intervals.

Nicolás Honorato-Droguett, Kazuhiro Kurita, Tesshu Hanaka et al.

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.