Monotonic Representations of Outerplanar Graphs as Edge Intersection Graphs of Paths on a Grid

A graph $G$ is called an edge intersection graph of paths on a grid if there is a grid and there is a set of paths on this grid, such that the vertices of $G$ correspond to the paths and two vertices of $G$ are adjacent if and only if the corresponding paths share a grid edge. Such a representation is called an EPG representation of $G$. $B_{k}$ is the class of graphs for which there exists an EPG representation where every path has at most $k$ bends. The bend number $b(G)$ of a graph $G$ is the smallest natural number $k$ for which $G$ belongs to $B_k$. $B_{k}^{m}$ is the subclass of $B_k$ containing all graphs for which there exists an EPG representation where every path has at most $k$ bends and is monotonic, i.e. it is ascending in both columns and rows. The monotonic bend number $b^m(G)$ of a graph $G$ is the smallest natural number $k$ for which $G$ belongs to $B_k^m$. Edge intersection graphs of paths on a grid were introduced by Golumbic, Lipshteyn and Stern in 2009 and a lot of research has been done on them since then. In this paper we deal with the monotonic bend number of outerplanar graphs. We show that $b^m(G)\leqslant 2$ holds for every outerplanar graph $G$. Moreover, we characterize in terms of forbidden subgraphs the maximal outerplanar graphs and the cacti with (monotonic) bend number equal to $0$, $1$ and $2$. As a consequence we show that for any maximal outerplanar graph and any cactus a (monotonic) EPG representation with the smallest possible number of bends can be constructed in a time which is polynomial in the number of vertices of the graph

Finding Optimal 2-Packing Sets on Arbitrary Graphs at Scale

A 2-packing set for an undirected graph $G=(V,E)$ is a subset $\mathcal{S} \subset V$ such that any two vertices $v_1,v_2 \in \mathcal{S}$ have no common neighbors. Finding a 2-packing set of maximum cardinality is a NP-hard problem. We develop a new approach to solve this problem on arbitrary graphs using its close relation to the independent set problem. Thereby, our algorithm red2pack uses new data reduction rules specific to the 2-packing set problem as well as a graph transformation. Our experiments show that we outperform the state-of-the-art for arbitrary graphs with respect to solution quality and also are able to compute solutions multiple orders of magnitude faster than previously possible. For example, we are able to solve 63% of our graphs to optimality in less than a second while the competitor for arbitrary graphs can only solve 5% of the graphs in the data set to optimality even with a 10 hour time limit. Moreover, our approach can solve a wide range of large instances that have previously been unsolved