On the bend-number of planar and outerplanar graphs
LATIN'12 Proceedings of the 10th Latin American international conference on Theoretical Informatics
Approximation algorithms for B 1-EPG graphs
WADS'13 Proceedings of the 13th international conference on Algorithms and Data Structures
Edge-intersection graphs of grid paths: The bend-number
Discrete Applied Mathematics
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We combine the known notion of the edge intersection graphs of paths in a tree with a VLSI grid layout model to introduce the edge intersection graphs of paths on a grid. Let 𝒫 be a collection of nontrivial simple paths on a grid 𝒢. We define the edge intersection graph EPG(𝒫) of 𝒫 to have vertices which correspond to the members of 𝒫, such that two vertices are adjacent in EPG(𝒫) if the corresponding paths in 𝒫 share an edge in 𝒢. An undirected graph G is called an edge intersection graph of paths on a grid (EPG) if G = EPG(𝒫) for some 𝒫 and 𝒢, and 〈𝒫,𝒢〉 is an EPG representation of G. We prove that every graph is an EPG graph. A turn of a path at a grid point is called a bend. We consider here EPG representations in which every path has at most a single bend, called B1-EPG representations and the corresponding graphs are called B1-EPG graphs. We prove that any tree is a B1-EPG graph. Moreover, we give a structural property that enables one to generate non B1-EPG graphs. Furthermore, we characterize the representation of cliques and chordless 4-cycles in B1-EPG graphs. We also prove that single bend paths on a grid have Strong Helly number 3. © 2009 Wiley Periodicals, Inc. NETWORKS, 2009