String graphs of k-bend paths on a grid

Andrei Asinowski, Elad Cohen, Martin Charles Golumbic, Vincent Limouzy, Marina Lipshteyn, Michal Stern

Research output: Contribution to journalArticlepeer-review


We investigate the class of vertex intersection graphs of paths on a grid, and specifically consider the subclasses that are obtained when each path in the representation has at most k bends (turns). We call such a subclass the Bk-VPG graphs, k≥0.We present a complete hierarchy of VPG graphs relating them to other known families of graphs. String graphs are equivalent to VPG graphs. The grid intersection graphs are shown to be equivalent to the bipartite B0-VPG graphs. Chordal B0-VPG graphs are shown to be exactly Strongly Chordal B0-VPG graphs. We prove the strict containment of B0-VPG and circle graphs into B1-VPG. Planar graphs are known to be in the class of string graphs, and we prove here that planar graphs are B3-VPG graphs.In the case of B0-VPG graphs, we observe that a set of horizontal and vertical segments have strong Helly number 2. We show that the coloring problem for Bk-VPG graphs, for k≥0, is NP-complete and give a 2-approximation algorithm for coloring B0-VPG graphs. Furthermore, we prove that triangle-free B0-VPG graphs are 4-colorable, and this is best possible.

Original languageEnglish
Pages (from-to)141-146
Number of pages6
JournalElectronic Notes in Discrete Mathematics
Issue numberC
StatePublished - 1 Aug 2011

Bibliographical note

Funding Information:
1 partially supported by the Israel Science Foundation grant 347/09


  • Chordal Graphs
  • Circle Graphs
  • Graph Coloring
  • Grid Intersection Graphs
  • Helly Property
  • Planar Graphs
  • String Graphs
  • Triangle-free Graphs

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Applied Mathematics


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