6 research outputs found
Visibility Representations of Boxes in 2.5 Dimensions
We initiate the study of 2.5D box visibility representations (2.5D-BR) where
vertices are mapped to 3D boxes having the bottom face in the plane and
edges are unobstructed lines of sight parallel to the - or -axis. We
prove that: Every complete bipartite graph admits a 2.5D-BR; The
complete graph admits a 2.5D-BR if and only if ; Every
graph with pathwidth at most admits a 2.5D-BR, which can be computed in
linear time. We then turn our attention to 2.5D grid box representations
(2.5D-GBR) which are 2.5D-BRs such that the bottom face of every box is a unit
square at integer coordinates. We show that an -vertex graph that admits a
2.5D-GBR has at most edges and this bound is tight. Finally,
we prove that deciding whether a given graph admits a 2.5D-GBR with a given
footprint is NP-complete. The footprint of a 2.5D-BR is the set of
bottom faces of the boxes in .Comment: Appears in the Proceedings of the 24th International Symposium on
Graph Drawing and Network Visualization (GD 2016
Line graphs of bounded clique-width
AbstractWe show that a set of graphs has bounded tree-width or bounded path-width if and only if the corresponding set of line graphs has bounded clique-width or bounded linear clique-width, respectively. This relationship implies some interesting algorithmic properties and re-proves already known results in a very simple way. It also shows that the minimization problem for NLC-width is NP-complete
Embeddings of k-Connected Graphs of Pathwidth k
We present O(n ) embedding algorithms (subgraph isomorphism and its generalizations) for classes of graphs of bounded pathwidth, where n is the number of vertices in the graph. These include the rst polynomial-time algorithm for minor containment and the rst O(n ) algorithm (c a constant independent of k) for topological embedding of graphs from subclasses of partial k-trees, as well as an O(n ) algorithm for subgraph isomorphism. Of independent interest are structural properties of k-connected graphs of bounded pathwidth on which our algorithms are based. We also describe special cases which reduce to various generalizations of string matching, permitting more ecient solutions. Finally, we describe n algorithms for solving these problems on any graphs of pathwidth at most k