Abstract
Edge-intersection graphs of paths in grids are graphs that can be represented such that vertices are paths in a grid and edges between vertices of the graph exist whenever two grid paths share a grid edge. This type of graphs is motivated by applications in conflict resolution of paths in grid networks. In this paper, we continue the study of edge-intersection graphs of paths in a grid, which was initiated by Golumbic, Lipshteyn and Stern. We show that for any k, if the number of bends in each path is restricted to be at most k, then not all graphs can be represented. Then we study some graph classes that can be represented with k-bend paths, for small k. We show that every planar graph has a representation with 5-bend paths, every outerplanar graph has a representation with 3-bend paths, and every planar bipartite graph has a representation with 2-bend paths. We also study line graphs, graphs of bounded pathwidth, and graphs with -regular edge orientations.
Highlights
Presume you have a network and need to route calls in it
Since calls interfere with each other, you need to route calls such that no two connections share a link. This transforms into a colouring problem in an edge-intersection graph of paths
The network is the host graph, and each call becomes a path in the host graph; calls share a link if and only if the paths share an edge of the host graph
Summary
Since calls interfere with each other, you need to route calls such that no two connections share a link. This transforms into a colouring problem in an edge-intersection graph of paths. Golumbic, Lipshteyn and Stern generalized the framework by allowing networks that are grids, rather than trees [10]. They define EPG graphs to be the edge-intersection graphs of paths in a grid. We study some other graph classes, such as line graphs, graphs of bounded pathwidth, and graphs that have edge orientations with bounded indegrees
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